Communications II LABORATORY : Lab1- Signal Statistics, an Introduction to Simulink and FM

Transcription

1 Communicaions II LABORATORY : Lab1- Signal Saisics, an Inroducion o Simulink and FM Inroducion: In oday's lab we have hree pars. Throughou he firs par we will develop ools for analyzing, modifying, processing and exracing informaion from signals mahemaically. One of he mos basic (and someimes mos useful) mehods involves he calculaion of signal saisics. Calculaing signals saisics provides us a subsanial amoun of useful informaion abou a signal. These saisics allow us o deermine how much signal is presen (i.e., he signal srengh), how long a signal lass, wha values he signal akes on, and so on. We will use signal saisics o develop measures of signal qualiy (wih respec o a reference signal). The second par gives you a quick glance o he fundamenals of Simulink. I alks abou he simulaion parameers and clarify each parameer. Also i inroduces you away o esimae he power specral densiy and bandwidh of a given signal. The hird par is an applicaion on Simulink. I gives you he abiliy o simulae an Analog communicaion sysem ha uses frequency modulaion (FM). I discusses he bandwidh and he waveform of he message hroughou baseband sage, modulaion sage and reconsrucion sage. Par I, Signal Saisics and Qualiy: Applicaion: you can easily perform signal deecion (by deermining when a signal conains useful informaion raher han jus background noise). The Quesion How can I quaniaively deermine a signal s qualiy? Background 1. Signal Saisics When dealing wih a signal, i is ofen useful o obain a rough sense of he range of values i akes and of he average size of is values. We do his by compuing one or more signal saisics. The following liss a number of common signal saisics. I gives he defining formula for each for boh coninuous-ime and discree-ime signals. Also included is MATLAB code for calculaing he saisic for a discree-ime signal. If we wish o compue a saisic for a coninuous-ime signal when we only have a sampled represenaion, we can use he discree-ime saisic o approximae he coninuous saisic. The formulas needed for his approximaion are included here wih he label sampled. (In mos cases, his approximaion becomes beer as he sampling inerval T s decreases.) For compleeness, signal suppor and duraion are also defined below. 1. Suppor Inerval. A signal s suppor inerval (also occasionally known as jus he (signal s suppor or is inerval) is he smalles inerval ha includes all non-zero values of he signal. Coninuous-ime: 1 Discree-ime: n 1 n n MATLAB: n=n 1 :n for a signal s. Page 1

2 . Duraion. The duraion of a signal is simply he lengh of he suppor inerval. Coninuous-ime: - 1 Discree-ime: n - n MATLAB: Assumed lengh(s) for a signal s Sampled: ( - 1 ) = (n - n 1 + 1)*T s 3. Periodiciy. The key formulas are included here. Coninuous-ime: s() =s( + T ) Discree-ime: s[n] =s[n + N] Sampled: T NT s 4. Maximum and Minimum Value. These values are he larges and smalles values ha a signal akes on over some inerval defined by n 1 and n. In MATLAB hese values are found using he min and max commands. MATLAB: Maximum(s) =max(s) MATLAB: Minimum(s) =min(s) 5. Average Value. The average value, M, is he value around which he signal is cenered over some inerval Coninuous ime Discree ime Malab Sampled M ( s( )) M ( s[ n]) n 1 1 n M ( s) mean( s) M ( s( )) M ( s[ n]) s( ) d 1 n n 1 s[ n]. 6. Mean-squared value. The mean-squared value (or MSV) of a signal, MS, is defined as he average squared valued of he signal over an inerval. The MSV is also called he average power, because he squared value of a signal is considered o be he insananeous power of he signal. Coninuous ime Discree ime Malab Sampled MS( s( )) MS( s[ n]) n 1 1 n MS( s) mean( s.^) MS( s( )) M ( s[ n]) s ( ) d 1 n n 1 s [ n] 7. Roo mean squared value. The roo mean squared value (or RMS value) of a signal over some inerval is simply he square roo of mean squared value. Coninuous ime RMS( s( )) s ( ) d Discree ime Malab Sampled RMS( s[ n]) 1 n 1 RMS( s) sqr( mean( s.^)) RMS( s( )) RM ( s[ n]) n 1 n n1 s [ n] Page

3 8. Signal Energy. The energy of a signal, E, indicaes he srengh of a signal is presen over some inerval. Noe ha energy equals he average power imes he lengh of he inerval. Coninuous ime Discree ime Malab Sampled E( s( )) E( s[ n]) 1 s ( ) d n n1 s [ n] E( s) sum( s.^) E( s( )) E( s[ n]) T 9. Signal Value Disribuion. The signal value disribuion is a plo indicaing he relaive frequency of occurrence of values in a signal. There is no closed-form definiion of he signal value disribuion, bu i can be approximaed using a hisogram. A hisogram couns he number of samples ha fall wihin paricular ranges, or bins. Noe ha he y-axis is effecively arbirary, and ha he coarseness of he approximaion is deermined by he number of hisogram bins ha are used. Figure 1 shows an example of a signal value disribuion and he hisogram approximaion o ha disribuion. MATLAB: his(s,num_bins); s Figure 1 Signal Value Disribuion PrelabQ1: Generae a vecor of 000 random numbers ha are normally disribued. Hin: ry randn( ) Esimae he signal value disribuion (use 50 bins). Measuring signal disorion and error Suppose ha we wish o ransmi a signal from one locaion o anoher. This is a common ask in communicaion sysems. A common problem is ha he signal is ofen modified or disored in he communicaion process. Thus, he received signal is no he same as he ransmied signal. Typically, we wan o reduce he amoun of disorion as much as possible. However, his requires ha we have a mehod of measuring he amoun of disorion in a signal. In order o develop such a measure, we ll look a a signal plus noise model of signal disorion. Suppose we are ransmiing a signal s[n] over FM radio. Someone unes in o our radio saion and receives a modified version of our signal, r[n]. We can represen his modificaion mahemaically as he addiion of an error signal, v[n], like his: r[ n] s[ n] v[ n] Assuming ha we have boh s[n] and r[n], we can easily calculae v[n] as Page 3

4 v[ n] r[ n] s[ n] Noe ha if s[n] and r[n] are idenical, v[n] will be zero for all n. This suggess ha we can simply measure he signal srengh of v[n] by using one of he energy or power saisics. Mean squared value is a naural choice because i normalizes he error wih respec o he lengh of he signal. Someimes, hough, he RMS value is more desirable because i produces error values ha are direcly comparable o he values in v[n]. When we measure he MSV of an error signal, we someimes call i he mean squared error or MSE. Similarly, he RMS value of an error signal is ofen called he roo mean squared error or RMSE. In MATLAB, we will usually wan o calculae he MSE or RMSE over he enire lengh of he signals ha we have. Supposing ha we are given a signal s and a modified version s_mod (wih he same size), we can calculae he MSE and RMSE like his: mse mean(( s s _ mod).^); rmse sqr( mean(( s s _ mod).^)); Noice ha we could also subrac s from s_mod, he order doesn maer because of he square operaion. Also noe ha you mus include he period before he exponeniaion operaor in order o correcly square each sample. PrelabQ: A simple signal and is saisics Use he following MATLAB commands o creae a signal: >> n = 1:50; >> s = sin(*pi*n/50); (a) Use sem o plo he signal. Make sure ha you include: The figure iself. An x-axis label and a y-axis label. A figure number and a capion ha describes he figure. (b) Calculae he following saisics over he lengh of he signal (i.e., le n 1 =1 and n = lengh(s)), and include your resuls in your repor. Maximum value Minimum value Mean value Mean squared value RMS value Energy (c) Suppose ha s is he resul of sampling a coninuous-ime signal wih a sampling inerval Ts = 1=100. Use he discree-ime saisics o esimae he following saisics for he coninuous-ime signal s() = sin(4π): Signal duraion Energy Average power RMS Value Page 4

5 Laboraory Assignmen 1. Saisics of real-world signals Download he file lab1_daa.ma from he W:\lab1uiliies\saisics. Place i in he presen working direcory or in a direcory on he pah, and ype >> load lab1_daa This file conains wo signals which will be loaded ino your workspace. You will use he signal clarine in his problem and in Problem. (a) Esimae he fundamenal period of clarine. Include your esimae for he discree-ime signal (in samples). Include your esimae for he original coninuous-ime signal (in seconds). (b) Use he his command o esimae he signal value disribuion of clarine. Use 50 bins. Include he figure (wih axis labels, code, ec.) in your repor. From he hisogram, make an educaed guess of he MSV and RMSV. Explain how you arrived a hese guesses. (c) Calculae he following (discree-ime) saisics over he lengh of he signal: Mean value Energy Mean squared value RMS value. Looking a and measuring signal disorion In his problem, we ll measure he amoun of disorion inroduced o a signal by wo sysems. Download he wo files lab1_sys1.m and lab1_sys.m. Apply each sysem o he variable clarine using he following commands: >> sys1_ou = lab1_sys1(clarine); >> sys_ou = lab1_sys(clarine); (a) (Examine he effecs of he sysems) Use plo and MATLAB s zoom capabiliies o display roughly one period of: The inpu and oupu of lab1_sys1 on he same figure. The inpu and oupu of lab1_sys on he same figure. (b) (Describe he effecs of he sysems) Wha happens o he signal when i is passed hrough hese wo sysems? Look a your plos from he previous secion and describe he effec of: lab1_sys1.m on clarine. lab1_sys.m on clarine. (c) (Measure he disorion) Calculae he RMS error inroduced by each sysem. RMS error inroduced by lab1_sys1. RMS error inroduced by lab1_sys. Which sysem inroduces he leas error? Is his wha you would have expeced from your plos? Page 5

6 Par II, Inroducion o Simulink: In his experimen and ohers o follow, we will use he Simulink exension o Malab. Simulink is an icon-driven dynamic simulaion package ha allows he user o represen a sysem or a process by a block diagram. By he erm dynamic sysems, we refer o sysems whose oupus change over ime. Once he represenaion is compleed, Simulink may be used o digially simulae he behavior of he coninuous or discree-ime sysem. Simulink inpus can be Malab variables from he workspace, or waveforms or sequences generaed by Simulink iself. These Simulink-generaed inpus can represen coninuous-ime or discree-ime sources. The behavior of he simulaed sysem can be moniored using Simulink's version of common lab insrumens: scopes, specrum analyzers and nework analyzers. The oupus of hese "devices" are displayed in graph windows. The oupu of a sysem can be viewed using one of he monioring devices lised above, or i may be saved o a variable creaed in he Malab workspace. Background: Power Specral Densiy (PSD) is he frequency response of a random or periodic signal. I ells us where he average power is disribued as a funcion of frequency. The PSD of a random ime signal x() can be expressed in one of wo ways ha are equivalen o each oher: The PSD is he average of he Fourier ransform magniude squared, over a large ime Inerval. T 1 Sx ( f ) lim E x( ) e T T T j The PSD is he Fourier ransform of he auo-correlaion funcion. Sx ( f ) Rx( ) e T R ( ) E x( ) x * j d ( ) Geing Sared wih Simulink In his secion, we will learn he basics of Simulink and build a simple sysem. x T d Figure Power Specral Densiy of sinusoidal signal In order o familiarize yourself wih Simulink, you will firs build he sysem shown in Fig.. This sysem consiss of a funcion generaor ha feeds a scope and a Specrum Analyzer block. Page 6

7 1. Open a window for a new sysem by using he New opion from he File pull-down menu, and selec Model.. Drag he Signal Generaor and Scope, from he simulink (library) sources and simulink (library) sinks, respecively ino he new window you creaed. 3. Copy he specrum analyzer block (W:\lab1uiliies\specrum_analyzer) ino your working communicaion folder. Drag he Specrum Analyzer block from he File open (pah of specrum analyzer block), his block sores up a buffer of he inpu poins and hen displays he ime domain represenaion along wih he frequency conen of he buffer in he graph window using FFT. 4. Now you need o connec hese hree blocks. Your sysem should look like Fig.. 5. Double click on he Scope block o make he ploing window for he scope appear. 6. Se he simulaion parameers by selecing Simulaion parameers from he Simulaion pulldown menu. Under he Solver ab, se he Sop ime o 50, and he Max sep size o 0.0. Then selec OK. This will allow he Power Specral Densiy block o make a more accurae calculaion. 7. Sar he simulaion by using he Sar opion from he Simulaion pull-down menu. 8. A sandard Malab figure window will pop up showing he oupu of he Specrum Analyzer. 9. Change he frequency of he sine wave o 5 π rad/sec by double clicking on he Sine Wave icon and changing he number in he Frequency. Resar he simulaion. 10. Observe he change in he waveform and is specral densiy. If you wan o change he ime scaling in he plo generaed by he specrum analyzer, from he Malab promp use he subplo(,1,1) and axis( ) commands. 11. To esimae he BW of he inpu signal, use subplo(,1,) and crosshair( )-you can find his funcion in W:\Lab1Uiliies\crosshair. 1. When you are done, close he sysem window you creaed by using he Close opion from he File pull-down menu. PrelabQ3: Skech he PSD of sin wave having ω c = 5 π rad/sec and compare i wih he oupu of he simulaion resul. A word abou he Simulaion Parameers Dialog Box: Simulaion Time You can change he sar ime and sop ime for he simulaion by enering new values in he Sar ime and Sop ime fields. Simulaion ime and acual clock ime are no he same. For example, running a simulaion for 10 seconds usually does no ake 10 seconds. The amoun of ime i akes o run a simulaion depends on many facors, including he model's complexiy, he solver's sep sizes, and he compuer's speed. Solvers Simulaion of a Simulink model enails compuing is inpus, oupus, and saes a inervals from he simulaion sar ime o he simulaion end ime. Simulink uses a solver o perform his ask. No one mehod for solving a model is suiable for all models. Simulink herefore provides an assormen of solvers, each geared o solving a specific ype of model. The Solver pane allows you o selec he solver mos suiable for your model Page 7

8 Fixed-sep coninuous solvers These solvers compue a model's coninuous saes a equally spaced ime seps from he simulaion sar ime o he simulaion sop ime. Variable-sep coninuous solvers These solvers decrease he simulaion sep size o increase accuracy when a sysem's coninuous saes are changing rapidly and increase he sep size o save simulaion ime when a sysem's saes are changing slowly. Sep Sizes For variable-sep solvers, you can se he maximum and suggesed iniial sep size parameers. Maximum sep size The Max sep size parameer conrols he larges ime sep he solver can ake. The defaul is deermined from he sar and sop imes. If he ime span of he simulaion is very long, he defaul sep size migh be oo large for he solver o find he soluion. Imporan Facs abou Simulink: Many block parameers are unable. A unable parameer is a parameer whose value can change while Simulink is execuing a model his is which called Malab has Tunable Parameers. Also you should noe ha he sample ime is no a unable parameer i.e. you canno change he Sample Time parameer of a block while a simulaion is running. If you wan o change a block's sample ime, you mus sop and resar he simulaion for he change o ake effec. Anoher imporan noe is ha Simulink uses separae windows o display a block library browser, a block library, a model, and graphical (scope) simulaion oupu. These windows are no MATLAB figure windows and canno be manipulaed using Handle Graphics commands i.e. you can no use he command axis( ), subplo( ),.ec o modify he simulaion oupu of a simulink block. Can you explain wha we have done in page7 sep 10? Page 8

9 Par III, Simulink and Frequency Modulaion (FM): The objecive of his secion is o increase he sudens familiariy wih boh simulink and frequency modulaion (FM) signals. Where we will do he following: Creae an FM signal by modulaing a message signal ono a carrier. Examine he specrum of he modulaed carrier. Background: Angle modulaion is a process in which he angle of he modulaing sinusoidal carrier wave is varied according o he baseband signal. In his mehod of modulaion, he ampliude of he carrier wave is mainained consan. An imporan feaure of angle modulaion is ha i can provide beer discriminaion agains noise and inerference han ampliude modulaion. However, his improvemen in performance is achieved a he expense of increased ransmission bandwidh; ha is, angle modulaion provides us wih a pracical means of exchange channel bandwidh for improved noise performance. Such a rade off is no possible wih ampliude modulaion, regardless of is form. The mos commonly used mehods o vary he angle of he carrier signal in accordance o he message are phase modulaion (PM) and frequency modulaion (FM). In our experimen we will consider FM. FM is ha form of angle modulaion in which he insananeous frequency f i () is varied linearly wih he message signal m() as shown by: f i ()=f c + K f m(), where kf denoes a scaling facor, limiing he maximum frequency deviaion of signal ω= kf f () max S( ) A cos( k m( ) d ) As you know, from communicaions I class, he heoreical bandwidh of FM signals is infiniy. Anyway, we can approximae he BW of he FM signal using Carson s rule, which is defined as where f is he peak frequency deviaion. B. W. FM ( f B. W. m( )) In his par you will benefi from he specrum analyzer block in displaying he frequency conens of he modulaed signal and hus o esimae he signal bandwidh. c f 0 PrelabQ4: a. Sae in clear poins he advanages and disadvanages of using FM. b. Wrie he mahemaical expressions and plo he waveforms of FM modulaed carriers given ha he modulaing signal is: 1) m()=sin(ω m * ) ) Square wave c. Plo he ampliude specrum of a one modulaed carrier i.e. m()= Sin(ω m * ) assuming ω m =π x 7.5 r/s and ω c =π x75 r/s. d. Calculae he bandwidh of FM modulaed carriers given ha he modulaing signal as in par (b), where ω m =π * 7.5 r/s and square wave period T c =1/7.5 s [Hin: assume ha he Bandwidh of he square wave is 3 * fc (fc = 1/ Tc)] Page 9

10 Laboraory Assignmen: Design problem: We inend o build a communicaion sysem ha has he following design specificaions Modulaion ype: Frequency modulaion Modulaing frequency: 7.5 Hz Carrier frequency: 75 Hz Modulaing signal: sine wave Ampliude of modulaing signal: Vpp Ampliude of carrier signal: Vpp Building Frequency Modulaion (FM) Model 1. Sar Simulink by yping simulink in he Malab workspace.. Open a new model window (File New Model). 3. Creae he following FM model. Figure 3 FM communicaion sysem 4. Se he parameers of he differen blocks as follows: Signal generaor Waveform: Sine Ampliude: 1 Frequency:.5 Unis: Herz Selec checkbox (inerpre vecor parameer as 1-D) FM modulaor passband Carrier frequency: 75 Iniial phase:0 Modulaion consan: 5 Symbol ime: Symbol inerval: Inf FM demodulaor Passband: Lowpass filer numeraor: [ ].* 0.01 Lowpass filer denominaor: [ ] Analog Filer design: Desired mehod: Buerworh Filer ype: Lowpass Filer order: 5 Passband edge frequency: 350 Specrum Analyzer Number of sample poins: 51 Sample period: Ploing period: 10 Simulaion parameers (Simulaion simulaion parameers), Selec solver ab a. Sar ime 0.0 Sop ime 10 b. Type: fixed sep Ode5 (Dormand-price) c. Fixed sep size:0.001 Mode: Auo Page 10

11 5. Run he simulaion and answer he following: Use he scope o display he inpu signal and he modulaed signal. From Specrum Analyzer window, deermine he Bandwidh of he modulaing signal (message). Use crosshair( ) Esimae he Bandwidh of he modulaed FM signal. Use crosshair( ) Wha is he purpose of he lowpass filer in our model? Is he reconsruced signal (he signal a he demodulaor oupu) he same as he inpu signal? Now if we change he inpu o a square wave, hen he frequency of he carrier signal will change back and forward beween wo differen frequencies. This is a ype of digial modulaion echniques called Binary Frequency Shif Keying (BFSK). In his mehod, he symbol 1 and 0 are disinguished from each oher by he frequency of he carrier (modulaed) signal. Here you are invied o simulae his sysem and don' worry abou he deails, laer we will sudy such a sysem in deph. 6. Change he inpu message from a sine wave o a square wave Display he signal afer he modulaor sage in ime domain. Esimae he B.W. of he modulaed signal. Use crosshair( ) Is he reconsruced signal (he signal a he demodulaor oupu) he same as he inpu signal? Imporan noes abou FM modulaor and demodulaor blocks: Typically, an appropriae Carrier frequency value is much higher han he highes frequency of he inpu signal. To avoid having o use a high carrier frequency and consequenly a high sampling rae, you can use baseband simulaion (FM Modulaor Baseband block) insead of passband simulaion. By he Nyquis sampling heorem, he reciprocal of he Sample ime parameer mus exceed wice he Carrier frequency parameer. In he course of demodulaing, he block uses a filer whose ransfer he lowpass filer numeraor describes funcion and Lowpass filer denominaor parameers, which are lised in order of descending powers of s. Page 11

12 Lab - Sampling, Quanizaion, and Pulse Code Modulaion (PCM) Inroducion: Alhough a significan porion of communicaion oday is in analog form, i is being replaced rapidly by digial communicaion. Wihin he nex decade mos of communicaion will become digial, wih analog communicaion playing a minor role. Today's lab may be viewed as a ransiion from analog o digial communicaions; i will consider he firs imporan sep in any digial communicaion sysem, ransforming he source informaion o a form ha is compaible wih a digial sysem. We will rea various aspecs of sampling, quanizaion (boh uniform and nonuniform), and pulse code modulaion (PCM). Finally, as an applicaion, we will build a complee digial sysem ha deals wih a speech signal o invesigae he effec of he sysem parameers on he qualiy of he reconsruced speech signal. Prelab 1. How many bis would be required o represen an analog signal wih values ranging from (-1) o 1 Vol if he resuling quanized signal is o have a resoluion of 0.15 V? (round o he neares bi). Assuming a maximum source coding daa rae of 50 kbis/sec, wha is he maximum signal bandwidh we can ransmi using PCM wih he number of bis found in quesion 1? Wha is he corresponding Nyquis rae? 3. For he resuls in quesions 1 and, compue he Signal o Quanizaion Noise Raio (SQNR) V max of he sysem assuming he message signal has a peak o average power raio of 3 db Pav and 14 db. 4. Derive he he SQNR for a sine wave in erm of he number of levels. 5. Use he simulink o draw he characerisic of a quanizer having he following I/O relaion: y x x x 0 x For he sysem shown in page 10, calculae heoreically: a. he firs 5 samples b. quanized samples c. PCM code words d. he decoded quanized samples for he PCM 7. Repea he above quesion if he sysem having a μ-law compandor (compressor & expander) wih μ=55. Sampling process: The sampling process is usually described in he ime domain. I is an operaion ha is basic o digial signal processing and digial communicaion. Through he use of he sampling process, an Page 1

13 analog signal is convered ino corresponding sequence of samples ha is usually spaced uniformly in ime i.e. discree in ime. Clearly, for such a procedure o have a pracical uiliy, i is necessary ha we choose he sampling rae properly so ha he sequence of samples uniquely defines he original analog signal. The sampling heorem is he basis for deermining he proper sampling rae for a given band-limied signal. I saes ha, A band-limied signal of finie energy, which has no frequency componen higher han W herz, is compleely described by specifying he values of he signal a insans of ime separaed by 1/W seconds (i.e. a a rae of W samples per second). The process of reconsrucing a coninuous-ime signal from is samples is also known as inerpolaion. In which we pass he sampled signal hrough an ideal low-pass filer of bandwidh W Hz. As you may noe, he use of a sampling rae higher han he Nyquis rae has a beneficial effec of easing he design of he reconsrucion filer used o recover he original signal from is sampled version Time-Division Muliplexing: The sampling heorem provides he basis for ransmiing he informaion conained in a bandlimied message signal as a sequence of samples. An imporan feaure of he sampling process is conservaion in ime. Tha is, he ransmission of he message samples engages he communicaion channel for only a fracion of he sampling inerval on a periodic basis, and in his way some of he ime inerval beween adjacen samples is cleared for use by oher independen message source on a ime-shared basis. We hereby obain a ime-division muliplex (TDM) sysem, which enable he join uilizaion of a common communicaion channel by allowing all signals o share he ransmission link, wih each signal conneced o he link for only a shor ime. The rae of he commuaor (elecronic swiching circui) a he ransmier side also obeys he sampling heorem: f s =N* *W, where N is he number of messages and W= max(w i ), i=1,, N. I is clear ha he use of TDM inroduces a bandwidh expansion facor N, because he scheme mus squeeze N samples derived from N independen message sources ino a ime slo equal o one sampling inerval. A he receiving end of he sysem, he received signal is applied o a pulse demodulaor which consiss of a decommuaor and a LPF. The decommuaor is in synchronizaion wih he commuaor in he ransmier. Quanizaion: A coninuous ime signal, such as voice, has a coninuous range of ampliudes and herefore is samples have a coninuous ampliude range i.e. hey are only discree in ime no in ampliude. In oher words, wihin he finie ampliude range of he signal, we find an infinie number of ampliude levels. I is no necessary in fac o ransmi he exac ampliude of he samples. Any human sense (he ear or he eye), as ulimae receiver, can deec only finie inensiy differences. This means ha he original coninuous ime signal may be approximaed by a signal consruced of discree ampliudes seleced on a minimum error basis from an available se. Clearly, if we assign he discree ampliude levels wih sufficienly close spacing we may ake he approximaed signal pracically indisinguishable from he original coninuous signal. Ampliude quanizaion is defined as he process of ransforming he sample ampliude m(nt s ) of a message signal m() a ime =nt s ino a discree ampliude v(nt s ) aken from a finie se of possible ampliudes. The exisence of a finie number of discree ampliude levels is a basic condiion of pulse code modulaion. So Quanizaion is an imporan sage in forming he PCM signal where he oupu of Page 13

14 he sampling process is quanized o provide a new represenaion ha is discree in boh ime and ampliude. Quanizaion can be of a uniform or nonuniform ype. In a uniform quanizer, he represenaion levels are uniformly spaced; oherwise, he quanizer is nonuniform. In sysem ha uses uniform quanizer, he quanizaion noise is he same for all signal magniude. Therefore, in uniform quanizaion, he SNR is worse a low level signals han for high level signals. In elephone sysems, i was found ha for mos voice communicaion channel, very low speech volumes predominae; 50% of he ime, he volage characerizing deeced speech energy is less han 1/4 of he rms volage. Large ampliude values are relaively rare; only 15% of he ime does he volage exceeds he rms value. Also as you may have noiced ha he quanizaion noise depends on he sep size, so a uniform quanizer would be waseful for speech signal. Nonuniform quanizaion, in which he sep size increases as he separaion from he origin of he inpuoupu ampliude increases, can provide fine quanizaion o he weak signals and coarse quanizaion of he srong signals. In oher words, he weak passages, which need more proecion, are favored a he expense of he loud passages. Thus in he case of nonuniform quanizaion, quanizaion noise can be made proporional o signal size. The effec is o improve he overall SNR by reducing he noise for he predominan weak signals, a he expense of an increase in noise for he rarely occurring signals. The use of non-uniform quanizer is equivalen o passing he baseband signal hrough a compressor and hen applying he compressed signal o a uniform quanizer. Quanizaion and Compression Quanizaion is someimes used for compression. As an example, suppose we have a digial image which is represened by 8 differen gray levels: [ ]. To direcly sore each of he image values, we need a leas 8-bis for each pixel since he values range from 0 o 55. However, since he image only akes on 8 differen values, we can assign a differen 3-bi ineger (a code) o represen each pixel: [ ]. Then, insead of soring he acual gray levels, we can sore he 3-bi code for each pixel. A look-up able, possibly sored a he beginning of he le, would be used o decode he image. This lowers he cos of an image considerably: less hard drive space is needed, and less bandwidh is required o ransmi he image (i.e. i downloads quicker). In pracice, here are much more sophisicaed mehods of quanizing images which rely on quanizaion. Pulse Code Modulaion Pulse Code Modulaion (hereinafer referred o as PCM) is a sampled modulaion similar o Pulse Ampliude Modulaion. Since PCM encodes a message ino bis of 1 s and 0 s, i is ofen referred o as a source code. PCM does no yield waveforms ha vary linearly wih he message however. Nyquis crieria apply o PCM since i is obained hrough sampling. Tha means ha he sampling frequency mus be a leas wice he highes frequency in he message. Page 14

15 PCM offers advanages over oher modulaion mehods in is resisance o noise and is abiliy o be processed digially. PCM is paricularly resisan o noise added once i has been modulaed in he communicaion channel. PCM can be processed enirely in he digial domain, allowing any desired signal aleraions o be performed ha would oherwise be impossible in he analog domain. An Analog o Digial or A/D converer is used o conver he coninuous message signal ino a series of digial numbers, wih each of he numbers represening a level of he quanized message signal. This sream of digial number is he PCM signal. Once he quanized signal is in he digial domain, i is easy o perform any signal processing desired by simply performing he appropriae mahemaical operaion. A D/A converer performs an inverse operaion o ha of he A/D converer, changing digial numbers o analog volages. The A/D and D/A converers mus mach wih respec o word size, sampling rae, and mapping in order for he PCM signal o be properly demodulaed. Page 15

16 Laboraory Assignmen 1. Image Quanizaion Download he file 'founainbw.if' from he W:\lab1uiliies\saisics. Place i in he presen working direcory or in a direcory on he pah. The image 'founainbw.if' is an 8-bi grayscale image. We will invesigae wha happens when we quanize i o smaller numbers of bis/pixel. a) Load i ino Malab and display i using he following sequence of commands. y = imread('founainbw.if'); image(y); colormap(gray(56)); axis('image'); b) The image array will iniially be of ype uin8, so you will need o conver he image marix o ype double before performing any compuaion. Use he command z=double(y) c) Uniform quanizer implemenaion: There is an easy way o uniformly quanize a signal. Le where X is he signal o be quanized, and N is he number of quanizaion levels. To force he daa o have a uniform quanizaion sep of, Subrac Min(X) from he daa and divide he resul by. Round he daa o he neares ineger Muliply he rounded daa by and add Min(X) o conver he daa back o is original scale. d) Wrie a Malab funcion Y = Uquan(X,N) which will uniformly quanize an inpu array X (eiher a vecor or a marix) o N discree levels. e) Use his funcion o quanize he founain image o 7 b/pel, 6, 5, 4, 3,, 1 b/pel, and observe he oupu images. Prin hard copies of only he 7, 4,, and 1 b/pel images, as well as he original. INLAB REPORT: 1. Describe he arifacs (errors) ha appear in he image as he number of bis is lowered?. Noe he number of b/pel a which he image qualiy noiceably deerioraes. 3. Hand in he prinous of he above four quanized images and he original. 4. Compare each of hese four quanized images o he original. Page 16

17 Laboraory Assignmen. Audio Quanizaion If an audio signal is o be coded, eiher for compression or for digial ransmission, i mus undergo some form of quanizaion. Mos ofen, a general echnique known as vecor quanizaion is employed for his ask, bu his echnique mus be ailored o he specific applicaion so i will no be addressed here. In his assignmen, we will observe he effec of uniformly quanizing he samples of wo audio signals. a) Download he file 'speech.au' and 'music.au' from he W:\lab1uiliies\saisics. Place i in he presen working direcory or in a direcory on he pah. b) Use your Uquan funcion o quanize each of hese signals o 7, 4, and 1 bis/sample. c) Lisen o he original and quanized signals and answer he following quesions: For each signal, describe he change in qualiy as he number of b/sample is reduced? For each signal, is here a poin a which he signal qualiy deerioraes drasically? A wha poin (if any) does i become incomprehensible? Which signal's qualiy deerioraes faser as he number of levels decreases? Do you hink 4 b/sample is accepable for elephone sysems?... b/sample? d) Use subplo o plo in he same figure, he four quanized speech signals over he index range 701:7400. e) Generae a similar figure for he music signal, using he same indices. Make sure o use orien all before prining hese ou. INLAB REPORT: Hand in answers o he above quesions, and he wo Malab figures. 3. Error Analysis As we have clearly observed, quanizaion produces errors in a signal. The mos effecive mehods of he analysis of he error signal urn ou o be probabilisic. In order o apply hese mehods, however, one needs o have a clear undersanding of he error signal's saisical properies. For example, can we assume ha he error signal is whie noise? Can we assume ha i is uncorrelaed wih he quanized signal? As you will see in his exercise, boh of hese are good assumpions if he quanizaion inervals are small compared wih sample-o-sample variaions in he signal. If he original signal is X, and he quanized signal is Y, he error signal is defined by he following: E = Y X When he spacing,, beween quanizaion levels is sufficienly small, a common saisical model for he error is a uniform disribuion from - / o /. a) Compue he error signal for he quanized speech for 7, 4, and 1 b/sample. b) Use he command his(e,0) o generae a 0-bin hisogram for each of he four error signals. Use subplo o place he four hisograms in he same figure. INLAB REPORT: Page 17

18 1. Hand in he hisogram figure.. How does he number of quanizaion levels seem o affec he shape of he disribuion? 3. Explain why he error hisograms you obain migh no be uniform? 4. Signal o Noise Raio One way o measure he qualiy of a quanized signal is by he Power Signal-o-Noise Raio (PSNR). This is defined by he raio of he power in he quanized speech o power in he noise. In his expression, he noise is he error signal E. Generally, his means ha a higher PSNR implies a less noisy signal. From previous labs we know he power of a sampled signal, x(n), is defined by where L is he lengh of x(n). Compue he PSNR for he four quanized speech signals from he previous secion. In evaluaing quanizaion (or compression) algorihms, a graph called a "rae-disorion curve" is ofen used. This curve plos signal disorion vs. bi rae. Here, we can measure he disorion by 1, and deermine he bi rae from he number of quanizaion levels and sampling rae. For PSNR example, if he sampling rae is 8000 samples/sec, and we are using 7 bis/sample, he bi rae is 56 kilobis/sec (kbps). Assuming ha he speech is sampled a 8kHz: Plo he rae disorion curve using 1 as he measure of disorion. Generae his curve by PSNR compuing he PSNR for 7, 6, 5,..., 1 bis/sample. Make sure he axes of he graph are in erms of disorion and bi rae. INLAB REPORT: Hand in a lis of he 4 PSNR values, and he rae-disorion curve. Page 18

19 Laboraory Assignmen Design problem: We inend o build a complee PCM communicaion sysem ha has he following design specificaions Message signal(x()): sine wave Sampling frequency: 100 Hz Number of bis/sample: Frequency of x() : 5HZ Ampliude of message signal: Vpp Quanizer ype: Uniform Building PCM encoder Model a) Copy he file 'PCM.mdl' from he W:\lab1uiliies\saisics. Place i in he presen working direcory or in a direcory on he pah. b) Connec he blocks o ge he following PCM model. c) Se he parameer of each block o mee our design specificaions in he above design problem. d) Visualize he message signal hroughou he differen sages in he above sysem. e) How can we reconsruc he original coninuous ime signal from he signal a he oupu of he quanizer decoder? f) Wha is he resuling SQNR of your 4 bi PCM sysem for he sine signal? How does your resul compare wih your heoreical calculaions? Explain why you hink here are differences. g) Change he inpu of he quanizer ino a speech signal which you have used in he previous laboraory assignmen. h) Lisen o he oupu wave file; do you hink b/sample is accepable for speech signals? Page 19

20 Simulaion of μ-law Quanizer Build a 4-bi non-uniform PCM sysem using a μ-law compressor and expander. Tes your sysem using he sine and audio waves from he previous secion as inpus. Pick a value of μ which you hink will work bes for he audio signal. a. Wha value of μ did you choose and why? b. Wha is he resuling SQNR for he sine and audio signals? Compare your resuls o he previous exercise. Have you gained any performance? For which message signal do you ge he larges performance gain and why? INLAB REPORT: 1. Include prinous of all SIMULINK block diagrams you designed.. Answer all quesions and back hem up wih resuls, analysis, heory, and any compuaions. 3. Wrie a paragraph abou quesions and confusion you have experienced in his par of he lab. Page 0

21 Lab 3- Deecion of Binary Signals in Gaussian Noise Inroducion: We will invesigae mached filers, which play an imporan role in applicaions such as digial communicaion, radar, sonar, ulrasound imaging, and many ohers. We will consider he digial communicaion applicaion, in which he objecive is o ransmi binary daa (0's and 1's) from one locaion o anoher. Mached filers are defined by heir impulse response, so he filer oupu is compued by a convoluion operaion. We will see how he sysem performance is affeced by ype of degradaion ha occurs in every real communicaion sysem: passage of he daa sream is affeced addiive noise. The lab will conclude wih a design projec in which you will diagnose he shorcomings of a given digial communicaion sysem, and hen improve he performance by designing proper mached filers. Background: m() Informaion Source & Inpu ransducer Sampler & Quanizer Source encoder Digial modulaion Channel Oupu ransducer Source decoder Digial demodulaion Figure 3.1 Block diagram of basic elemens of a digial communicaion sysem Noise in Radio Communicaions Sysems The ask of he demodulaor or deecor is o rerieve he bi sream from he received waveform, as nearly error free as possible, nowihsanding he disorion o which he signal may have been subjeced. There are wo primary causes for signal disorion. The firs is filering effecs of he ransmier, channel, and receiver disorion. A non ideal sysem ransfer funcion causes symbol "smearing", which can produce inersymbol inerference. The second cause for signal disorion is he noise ha is produced by a variey of sources, such as galaxy noise, erresrial noise, amplifier noise, and unwaned signals from oher sources. An unavoidable cause of noise is he hermal moion of elecrons in a conducing media. This moion produces hermal noise in amplifiers and circuis which corrups he signal in addiive fashion; ha is, he received signal, r() is he sum of he ransmied signal, s() and he hermal noise, n(). The saisics of hermal noise have been developed using quanum mechanics and are well known. The primary saisical characerisic of hermal noise is ha he noise ampliudes are disribued according o a normal or Gaussian disribuion. The primary specral characerisic of hermal noise is ha is wo-sided power specral densiy, G(f)=N 0 /, is fla for all frequencies of ineres for radio communicaion sysems. In oher words, Page 1

22 hermal noise, on he average, has jus as much power per herz in low-frequency flucuaions as I 1 high-frequency flucuaions- up o a frequency of abou 10 herz. The adjecive "whie" is used in he sense ha whie ligh conains equal amoun of all frequencies wihin he visible ligh band of elecromagneic radiaion. Maximum Likehood Receiver Srucure: In a basic digial communicaion sysem, he opimum receiver for an AWGN channel is composed of wo pars; one is eiher a signal correlaor or a mached filer and he oher is a deecor. This lab considers a mached filer. The mached filer is assumed o be mached o he ransmied signal in his lab. Correlaor The signal correlaor cross correlaes he received signal wih all possible ransmied signals. Le us assume ha a M-ary scheme is used in a baseband digial communicaion sysem. The oupus of he signal correlaor, r o,, rm 1 in Figure 3., are as follows where () s i ( ), r ( ) r( ) s0 ( d 0 ) 0... M 1 ) r( ) sm 1 ( ) 0 r ( d. r is he received signal a he receiver and he possible ransmied signals are i 0,.., M 1, 0 T Afer we sample he correlaor oupus a ime =T, send hem direcly o he deecor. For example, le us consider ha here are wo possible ransmied signals in an AWGN channel. Two possible ransmied signals are assumed o be orhogonal each oher. s o () is assumed o be ransmied hrough an AWGN channel. The signal correlaor compues wo oupu signals as follows r( ) so ( ) n( ) r o T 0 s o T ( ) d s ( ) n( ) d E n T T 1 so ) s1( ) d so ( ) n( ) 0 0 r 0 o ( d n where r () is he received signal a he receiver. r o and r 1 are he correlaor oupus. n o and n 1 are he noise componens a he oupu of he correlaor. E is he energy of he signal, s o (). 1 o Page

23 s 0 ( ) =T r() d 0... =T r o Deecor s M 1 ( ) d 0 r M 1 Fig 3. Signal correlaor ; cross correlaing he received signal wih M ransmied signals Mached Filer The mached filer provides a mehod similar o he signal correlaor o demodulae he received signal r (). When a M-ary scheme is assumed o be used in a baseband digial communicaion sysem, he block diagram of mached filer is shown in Figure 3.3. The impulse response of he filer mached o he ransmied signal s () is represened by h( ) s( T ), 0 T When we assume he signal s () is ransmied, he oupu of he mached filer is as follows If we sample his oupu a =T, i is given by y( ) s( ) s( T ) d y( T) s ( ) d E 0 where E is he energy of he signal s (). 0 T Thus, he oupu of he mached filer sampled a =T is equal o he oupu of he signal correlaor. =T r() s0 ( T ) r o... =T Deecor s M 1( T ) r M 1 Figure 3.3 Mached Filer ; mached o he possible M received signals. Page 3

24 Simulaion Procedure: The simulaion model for he sysem o be simulaed is illusraed in Fig-3.4 Daa Generaion: The firs sep is o simulae he digial source. The source produces digial messages ha are discree in ime, have eiher 0 or 1 as one sample value and have a finie number of possible oupus. The messages are called a bi sequence having a sequence wih eiher 0 or 1 value. In he following code we will generae daa consising of a 1-by-N elemen in daa consiss of eiher 0 or 1. Txdaa = rand(1,10)>0.5 Fig 3.4 Simulaion Diagram model Encoder This block convers random inpu bis ino symbols ha are suiable for he ype of modulaion o be used. UpSampler and Transmi Filer. This block convers he inpu binary symbols ino a baseband waveform o be ransmied over he channel. The shape of he filer will deermine he shape of symbol ha represens he binary daa. We will convolve he maching filer wih he impulse rain o ge he daa shaped; ake a look o he figure below: Figure 3.5 Convoluion wih impulse rain Page 4

25 We need o reshape he encoded daa ino an impulse rain; his can be done wih he operaion called oversampling. Look o he following examples: X=[ ] % lengh =6 (X)= [ ] %lengh = *6=1 and insers -1 =1 zeros 3 (X)= [ ] %lengh=3*6=18, insers 3-1= zeros In Malab, o oversample he signal x by fs, we may use he following code: Oversamp=zeros(1,fs*lengh(x)) Oversamp(1:fs:end)=x Radio Communicaion Channel- AWGN If we consruc a mahemaical model for he signal a he inpu of he receiver, he channel is assumed o corrup he signal by addiion of noise. r()=s()+n() where s() is he ransmied signal, r() is he received signal and n() is a sample funcion of AWGN process wih probabiliy densiy funcion (pdf) and power specral densiy as follows: ( f ) 0.5NoW / Hz where No is a consan and ofen called he noise power densiy. To simulae in Malab, we simply use he buil-in funcion randn, which generaes random numbers and marices whose elemens are normally disribued wih mean 0 and variance 1. Therefore, if we add AWGN noise wih power 1 o he digial modulaion signal (Txdaa) daa_ch=txdaa+randn(1,lengh(txdaa)) However, in he simulaion, we someimes calculae he BER, performance by varying he noise power, where we define he noise power as a variable, npower. Since daa_ch is a volage, no power, we mus change he noaion npower from power o volage. We define a variable an as he oo of npower as: An = npower And hus he new Gaussian noise will be generaed as Noise= An* randn(1,n) Receive Filer and Sampler. This block convers he received baseband waveform o a symbol sream, which can be used a decision variable for decoding he received bis. In he firs case, he receive filer coefficiens are hose of an ideal inegraor (i.e, same as he ransmi filer coefficiens). This block also samples he received waveform. The sampling poin is a = ntb. Page 5

26 The sampled oupu is hen compared wih a predeermined hreshold o decide and hus guess he inpu daa sream. Bi Error Rae: We analyze he bi rae o evaluae he sysem performance. To calculae he number of errors, we subrac he ransmied signal from he received daa (afer making a decision). If no error exiss, you obain a zero vecor wih he lengh of daa. Oherwise, you obain a nonzero vecor in which "-1" or "1" daa occurs a he error posiions. The following code compues he number of errors occurred. Subdaa=rxdaa-xdaa NoE=sum(abs(Subdaa)) Page 6

27 Laboraory assignmen: Wrie a simulaion program o deermine he error probabiliy for a binary anipodal signaling sysem using he following signaling pulse shape, assuming a signaling rae of one bi per second (i.e. T = 1): s( ) E / T sin( / T), 0 T where E is he signal energy and 1=T is he bi-rae. For he purposes of simulaion, assume (wih no loss of generaliy) a normalized signaling rae of 1 signal/second (T=1), and a sampling rae (for programming purposes) of 0 samples/second (i.e., he ransmied pulse is represened by 0 uniformly spaced samples in he inerval from 0 o 1.) Plo he probabiliy of error as a funcion of he SNR in he range from 0 o 8 db in seps of 1 db under perfec ime synchronizaion a he oupu of he mached filer. Then, modify your simulaion program o obain he performance of he same sysem when here is a synchronizaion error of 15% and 30%. Plo all hree curves (0%, 15% and 30% synchronizaion error) in he same graph. Include in he same graph a fourh plo for he analyical probabiliy of error under perfec synchronizaion. Presen your resuls in he form of a shor repor and aach your Malab program in he and in an appendix. Include commens in your Malab program. Change he signaling shape ino s()=1 for 0 T and answer he requiremens above. Hins: 1. A "mached filer" implemenaion may be simpler (use he filer command of Malab).. To vary he SNR, i is bes o fix he signal energy E o 1 and hen vary he variance of he addiive noise. You can do his by generaing uni-variance Gaussian variables and muliplying hem by he appropriae sandard deviaion. Noe ha he SNR is defined as: E SNR where E is he signal energy (assumed o be 1) and is he variance of he addiive noise. 3. You should noe ha he performance of he sysem depends sricly on Eb/N0 raio so changing he shape of signaling, wih he same signal energy, does no change Page 7

28 Real Time Implemenaion of he Shaping and Mached Filer Using TIC6000 DSK Objecive: In his Lab we will use he TIC600 DSK o shape a binary sream of daa ino a binary PSK wih a sine shape. Experimenal Procedure: The firs sep is o find a suiable sampling frequency o represen he shaping filer wih a finie number of coefficiens. Thus he firs requiremen is o esimae he BW of he coninuous ime shaping filer. h( ) E / T sin(. / T), 0 T As he filer is finie in he ime domain, heoreically he bandwidh is infinie. Bu due o pracical issues, we can approximae he BW as he band ha conains he majoriy of he power of he filer. To esimae he bandwidh, we should look o he filer in he frequency domain. o find he Fourier ransform we may consider he filer as a produc of a sine wave wih a period *T and a rec signal wih a duraion of *T. Approximaely he B.W will be 3/T. (In your lab repor, prove i). As he sampling frequency of our DSK is consan and has a value of 8 KHz, we should have a filer wih a maximum BW of 4 KHz => T= ¾ msec => he maximum bi rae is 4/3 Kbi/sec using our DSK. To be in he safe side, ake T=1/500 sec. Afer deermining he bie ime (T), and knowing he sampling frequency fs=8 KHz (he same as he DSK), i is easy know o find he coefficien of shaping filer using Malab. Wrie a code in Malab o generae he coefficiens of he shaping filer. Use FIR_cof_gen, which we used in he DSP Lab, o wrie he coefficiens in a file named Shaping.coff. Keep in mind ha he funcion FIR_cof_gen scales he coefficien such ha i prevens overriding a he DSP processor which is no he case in his applicaion. So modify he funcion such ha i doesn scale he coefficiens. Using CCS wrie an Inerrup driven C code ha do he following: Sore a sream of binary daa (0 s and 1 s) in a marix named daa. Include he file ha conains he coefficiens of he shaping filer Read he sored binary daa number by number and hen shape each number wih he appreciae shape sin(. / T), 0 T Connec he oscilloscope o he oupu of he D/A and invesigae he oupu. Try o implemen he Mached filer in he same manner as above, bu you should pay a special aenion o a suiable synchronizaion approach. Page 8

29 Objecive Lab 4 - Performance of M-ary PAM and M-ary PSK Modulaion schemes In his par, our objecive is o sudy digial modulaion schemes in deail. We will consider differen mehods of digial modulaions: a binary pulse ampliude modulaion (PAM), a M-ary PAM, a binary phase shif keying (PSK), and a M-ary PSK. PULSE AMLITUDE MODULATION (PAM) In a pulse ampliude modulaion (PAM), he message bi sequences are modulaed wih carrier signals having differen ampliudes. When we ry o ransmi symbols having jus wo differen ampliude levels, i is called a binary PAM. When we ransmi symbols having M differen ampliude levels, i is called a M-ary PAM. BINARY PAM (PULSE AMLITUDE MODULATION) When we consider a binary PAM, we ransmi symbols having wo differen ampliude levels. In a baseband binary PAM case, he symbols are represened as follows s1( ) g( ), s ( ) g( ) where g () is a pulse shaping funcion ha could be a recangular or a raised cosine ec. If g () is a recangular; s ( ) 1 s ( ) A -A In a binary PAM, each symbol is used o ransmi 1 bi. If he symbol duraion is T, he ransmission rae is equal o 1/T bis/sec. M-ary PAM (PULSE AMLITUDE MODULATION) When we consider a baseband M-ary PAM, M symbols ha we wan o ransmi can be represened as follows sm ( ) Am g( ), m 1,,.., M A m ( m 1 M) d, d is he disance beween adjacen symbol ampliudes, and () where pulse shaping funcion ha can be a recangular pulse or a raised cosine and so on. g is he Assumed ha M is he number of differen symbols ha we can ransmi, may have he values as follows A m in he above equaion Page 9

30 I has he consellaion poins as below A ( M 1) d ( M 1 d * * * -(M-1)d -(M-3)d (M-1)d A A... A 1 3 ( M 3) d ( M 5) d M ) Figure Consellaion of a M-ary PAM in a signal space Simply, when we consider he carrier modulaed PAM, he modulaed signal may be represened as follows where T is a symbol duraion. s m ( ) Re{ A g( )exp( jf )} A g( )cos(f ), m m c c 0 T k When we assume ha symbols having M differen ampliude levels are ransmied, M is equal o. I means ha each symbol is used o ransmi k bis. If he symbol duraion is T, he ransmission rae is equal o k/t bis/sec. PHASE SHIFT KEYING (PSK) In a phase shif keying (PSK), he message bi sequences are modulaed wih carrier signals having differen phases. When we ry o ransmi symbols having jus wo differen phases, i is called a binary PSK. When we ransmi symbols having M differen phases, i is called a M-ary PSK. BINARY PSK (PHASE SHIFT KEYING) When we consider a binary PSK, we ransmi symbols having wo differen phases. In carrier modulaed sysems, s m ( ) Re g( )exp( j ( m 1) / M)exp( jf ) where m 1,, 0 T. ( m 1) g( )cos f c M c. The message informaion is embedded in he phase of he carrier signal, represened as follows ( m 1) ( m 1) sm ( ) g( )cos cos f c g( )sin sin f c M M where m=1,. m ( m 1) M and can be Page 30

31 Page 31 On he righ side of he above equaion, he firs erm is an inphase componen and he second erm is a quadraure componen of he low-pass equivalen of ) ( s m. The basis funcions of his are given by f g f f g f c g c g )sin ( ) ( ) cos ( ) ( 1. M m M m s g g m 1) ( sin 1) ( cos. In a binary PSK, M is equal o. Thus, he possible carrier phases are 0 and. A binary PSK is equivalen o a binary PAM. M-ary PSK (PHASE SHIFT KEYING) When we consider a M-ary PSK, we ransmi symbols having M differen phases. In carrier modulaed sysems, ) )exp( 1) / ( )exp( ( Re ) ( f j M m j g s c m M m f g c 1) ( )cos (. where T M m 0, 1,..,. The message informaion is embedded in he phase of he carrier signal, M m m 1) ( and can be represened as follows f M m g f M m g s c c m sin 1) ( )sin ( cos 1) ( )cos ( ) ( where m=1,,m. On he righ side of he above equaion, he firs erm is an inphase componen and he second erm is a quadraure componen of he low-pass equivalen of ) ( s m. The basis funcions of his are given by f g f f g f c g c g )sin ( ) ( ) cos ( ) ( 1. M m M m s g g m 1) ( sin 1) ( cos.

32 When we assume ha M is equal o 4, he possible carrier phases are 0, /,,3 /. In 8 PSK, he possible carrier phases are 0, / 4, /,...,7 / 4. The following figures show a signal space diagram for each M-ary PSK. g / g / (a) M = (b) M = 4 g / (c) M = 8 Figure 4..1 Consellaions of several M-ary PSK in a signal space; (a) a binary PSK, (b) 4 PSK, and (c) 8 PSK The minimum disance beween adjacen consellaion poins is d e min g (1 cos( / M )). OPTIMUM RECEIVER THROUGH AWGN CHANNEL In a basic digial communicaion sysem, he opimum receiver for an AWGN channel is composed of wo pars; one is eiher a signal correlaor or a mached filer and he oher is a deecor. We will consider a signal correlaor and a mached filer in he following wo secions. A maximum likelihood decoding algorihm will be explained in he hird secion and he performance for differen modulaion schemes will be evaluaed wih regard o he averaged probabiliy of error in he las secion. PERFORMANCE FOR AWGN CHANNEL The probabiliy of error can be a good measure for he performance of he modulaion scheme for AWGN channel. Mone Carlo simulaion is used o esimae and plo he probabiliies of eiher bi or symbol error for he performance of a digial communicaion sysem. Performance of a quadraure PSK Le us generae he message bi sequences having a lengh of digi. A quadraure PSK digial communicaion sysem is assumed in an AWGN channel. Use Mone Carlo simulaion o esimae and plo he performance of a quadraure PSK communicaion sysem reliably. Page 3

33 a) Plo he bi error probabiliy performance, heoreical and simulaed, versus he SNR per bi (0 ~ 10 db). Wha is he bi error probabiliy a 5dB? b) Plo he symbol error probabiliy performance, heoreical and simulaed, versus he SNR per bi (0 ~ 10 db). Wha is he symbol error probabiliy a 5dB? c) Plo he power specral densiy for he PSK signal. Performance of a 8-PSK digial Le us generae he message bi sequences having a lengh of digi. An 8-PSK digial communicaion sysem is assumed in an AWGN channel. Use Mone Carlo simulaion o esimae and plo he performance of an 8-PSK communicaion sysem reliably. a) Plo he bi error probabiliy performance, heoreical and simulaed, versus he SNR per bi (0 ~ 1 db). Wha is he bi error probabiliy a 5dB and 1dB? b) Plo he symbol error probabiliy performance, heoreical and simulaed, versus he SNR per bi (0 ~ 1 db). Wha is he symbol error probabiliy a 5dB and 1dB? c) Plo he power specral densiy for he PSK signal. Performance of a 4-PAM Le us generae he message bi sequences having a lengh of digis. A 4-PAM digial communicaion sysem is assumed in an AWGN channel. Use Mone Carlo simulaion o esimae and plo he performance of a 4-PAM communicaion sysem reliably. a) Plo he bi error probabiliy performance, heoreical and simulaed, versus he SNR per bi (0 ~ 1 db). Wha is he bi error probabiliy a 5dB? b) Plo he symbol error probabiliy performance, heoreical and simulaed, versus he SNR per bi (0 ~ 1 db). Wha is he symbol error probabiliy a 5dB? c) Plo he power specral densiy for he PAM signal. Performance of a 8-PAM Le us generae he message bi sequences having a lengh of digis. An 8-PAM digial communicaion sysem is assumed in an AWGN channel. Use Mone Carlo simulaion o esimae and plo he performance of an 8-PAM communicaion sysem reliably. a) Plo he bi error probabiliy performance, heoreical and simulaed, versus he SNR per bi (0 ~ 1 db). Wha is he bi error probabiliy a 1dB? b) Plo he symbol error probabiliy performance, heoreical and simulaed, versus he SNR per bi (0~1 db).wha is he symbol error probabiliy a 1dB? c) Plo he power specral densiy for he PAM signal. Page 33

34 Lab 5- Sudying he Performance of Digial Communicaion sysems Using Simulink Objecive: In his Lab we will inroduce he simulink as powerful ool o simulaing he performance of digial modulaion sysems wih ime. Also we will ake an easy example o sudy 16-Ary PSK using he communicaion oolbox funcions. Par One: Passband Simulaion This secion inroduces a passband simulaion model ha shows he error rae of QPSK modulaion over an AWGN channel wih a varying noise level. Wha Is he Communicaions Blockse? The Communicaions Blockse is a collecion of Simulink blocks for designing and simulaing communicaion sysems. Wih he Communicaions Blockse, you can design models in he form of block diagrams, using simple click-and-drag mouse operaions. You can run simulaions on a model a he push of a buon, and change parameers while he simulaion is running. The Communicaions Blockse conains ready-o-use blocks o model various processes wihin communicaion sysems, including Signal generaion Source coding Error-correcion Inerleaving Modulaion/demodulaion Transmission along a channel Synchronizaion In addiion, you can creae specialized blocks, o implemen your own algorihms. All he power of MATLAB is available o you when you use he Communicaions Blockse. You can run simulaions from he command line and invoke MATLAB expressions and M-files. Speed of Baseband versus Passband Models The passband and baseband models produce error raes ha differ from each oher by less han 1%. However, he passband model akes a significanly longer ime o process he same amoun of daa. Alhough he acual speed depends on your sysem, he relaive imes in he ables below can serve as a guide. Noice hese general rends: Baseband simulaion is considerably faser han passband simulaion. The difference in speed is especially dramaic when he carrier frequency in he passband simulaion is high. Baseband simulaion using large frames is faser han baseband simulaion ha does no use frames. Page 34

35 Laboraory Assignmen: We inend o build a digial communicaion sysem ha has he following design specificaions Modulaion ype: Passband PSK Carrier frequency: 30000Hz Bi rae: 18K bis/sec Symbol rae: 3K symbols/sec. Channel noise: AWGN Shaping filer: recangle, and ake18 samples o represen i. Delays in Digial Modulaion Digial modulaion and demodulaion blocks someimes incur delays beween heir inpus and oupus, depending on heir configuraion and on properies of heir signals. The following able liss sources of delay and he siuaions in which hey occur. As a resul of delays, daa ha eners a modulaion or demodulaion block a ime T appears in he oupu a ime T+delay. In paricular, if your simulaion compues error saisics or compares ransmied wih received daa, hen i mus ake he delay ino accoun when performing such compuaions or comparisons. In consrucing his model, o se he parameers for he Error Rae Calculaion block and Delay block correcly, you need o know he delay beween he ransmied and received signals. You can someimes deermine his delay from he parameers of he blocks beween he ransmied and received signals. Bu if you are unable o deermine he delay in his way, you can do so wih he xcorr funcion, from he Signal Processing Toolbox, which finds he cross correlaion beween he signal and shifs of is delayed version. To use he xcorr funcion, you mus modify he model slighly as follows: Se he Es/No parameer of he AWGN Channel block o 100. This essenially removes all noise from he model. Drag a Signal To Workspace block, from he DSP Sinks library, ino he model window. Connec he line leading ou of he random Ineger Generaor block o he Signal To Workspace block. Double-click he Signal To Workspace block o open is mask, and change he Variable name parameer o Tx. Drag anoher Signal To Workspace block, from he DSP Sinks library, ino he model window and connec i o he line leading ou of he M-PSK Demodulaor Passband block. Page 35

36 Double-click he second Signal To Workspace block ino he model window and change he Variable name parameer o Rx. Pull down he Simulaion menu and selec Simulaion parameers. Se Sop ime o 0.1. Deermining he Delay in he PSK Model Running he model sends he ransmied and received signals o he workspace as vecors called Tx and Rx, respecively. To find he delay beween Tx and Rx, ype he following commands a he MATLAB promp. [m,index]=max(xcorr(rx,tx)); L=lengh(Tx); delay=index-l To check he model we will run he model a no noise or le us say he ES/N0 is 100 db. See he error vecor. Se he parameers of he error rae calculaor as follow: Receive delay: he value found above Compuaional delay: 0. Targe number of errors: 100. Maximum number of symbols: Check he error vecor ErrorVec a he workspace and you will find i of lengh 3; he firs column is he probabiliy of error, he second is he oal number of errors found, and he hird is he oal number of daa compared. Wha values will his vecor have? Generaing Error Curves To plo error raes as a funcion of noise level using he simulaion, execue he following code in MATLAB. The plo includes boh he heoreical error raes and he error raes from running he simulaion. Change he value of Es/N0 a AWGN block o a variable called noise. We will link wih he Simulink hrough he workspace, so our scrip will modify he value of he variable noise which will subsequenly change he Es/No in our model. db=0:8; %Eb/N0, Bi energy in db. dbs=db+10*log10() % conver o symbol energy Es=log(M)*Eb, ake he logarihm in db for i=1:lengh(db) noise=dbs(i); % as we need o vary Es/N0 in our model no Eb/N0 sim('model name'); pe(i)=errorvec(1); end semilogy(db,pe) % plo he probabiliy versus he Ebi/N0. % Compuing heoreical values. linebnovec = 10.^ (0.1.* EbNoVec); emp = 0.5.*erfc((sqr(.*linEbNoVec))./sqr()); ser = emp.*( - emp); % Compues he heoreical symbol error hold on; semilogy(db,ser,'*') %plo he probabiliy versus he Ebi/N0. Page 36

37 Using Communicaion oolbox Simple Digial Modulaion Example This example illusraes he basic forma of he baseband modulaion and demodulaion commands, dmodce and ddemodce. Alhough he example uses he PSK mehod, mos elemens of his example apply o digial modulaion echniques oher han PSK. The example generaes a random digial signal, modulaes i, and adds noise. Then i creaes a scaer plo, demodulaes he noisy signal, and compues he symbol error rae. The ddemodce funcion demodulaes he analog signal y and hen demaps o produce he digial signal z. Noice ha he scaer plo does no look exacly like a signal consellaion. Whereas he signal consellaion would have 16 precisely locaed poins, he noise causes he scaer plo o have a small cluser of poins approximaely where each consellaion poin would be. However, he noise is sufficienly small ha he signal can be recovered perfecly. Noe Because some opions vary by mehod, you should check he reference pages before adaping he code here for oher uses. Below are he code and he scaer plo. M = 16; % Use 16-ary modulaion. Fd = 1; % Assume he original message is sampled % a a rae of 1 sample per second. Fs = 3; % The modulaed signal will be sampled % a a rae of 3 samples per second. % Use M-ary PSK modulaion o produce y. x = randin(100,1,m); % Random digial message y = dmodce(x,fd,fs,'psk',m); % Add some Gaussian noise. % Creae scaer plo from noisy daa. ynoisy = y +.04*randn(300,1) +.04*j*randn(300,1); scaerplo(ynoisy,1,0,'b.'); % Demodulae y o recover he message. z = ddemodce(ynoisy,fd,fs,'psk',m); s = symerr(x,z) % Check symbol error rae. Page 37

38 Lab 6- INTRODUCTION TO ERROR-CONTROL CODING Inroducion: In digial elecronic sysems, informaion is carried by signals wih discree ampliudes so ha he individual message bis can be more easily disinguished from one anoher. Bu when his digial informaion needs o be sored or ransmied over long disances or a very high speeds, noise becomes an issue. Sources of such unwaned noise include he hermal moion of elecrons, damage o he sorage media, or coupling from oher energy sources. In communicaions sysems, one way o decrease he probabiliy of bi errors is o increase he power of he ransmied signal unil i is much higher han he noise. Bu he amoun by which he signal power can be increased may be limied by he raing of he elecronic circuis in he ransmier and even a he receiving end. This power level may also be regulaed, as in he case of radio signals whose levels are specified by he Federal Communicaions Commission (FCC) in he Unied Saes. So clearly we need some oher means of conrolling he probabiliy of error. Forward error correcion, or channel coding, provides his added dimension. By adding redundan symbols o he ransmied or sored digial informaion, we can achieve no only a means of error deecion, bu error correcion as well. In his lab, we will uilize one such error conrol code, known as he (7,4) Hamming block code. (7,4) means ha 4 message bis, represened by he vecor m, are convered ino a 7-bi code, represened by he vecor x, according o he following able: This coding process can be convenienly represened in marix noaion (even hough alernaive mehods are ofen used in physical elecronic implemenaions): x = m G (modulo ) Page 38

39 Where G is he "generaor marix" given by: (he G, H and E marices used here are given by Simon Haykin, Digial Communicaions, Wiley, 1988, pp. 378f). Afer he message bis pass hrough a noisy medium ha inroduces unknown errors, we can model he received vecor y as follows: y = x + e (modulo ) In modulo addiion we have 0+0 = 1+1 = 0; 0+1 = 1+0 = 1. Therefore we imagine he error vecor e as a binary vecor; where a componen of e is 1, he received y vecor differs from he originaing x vecor. The receiver hen needs o decode he daa in an aemp o recover he original message bis. This produces a "syndrome", s, which deermines wha he error vecor was (if no errors or only a single bi error has occurred). This is again convenienly represened in marix noaion, hough physical elecronics may implemen i differenly: s = y H T (modulo ) Where H T is he ranspose of he "pariy-check" marix, H, given by: The 3-bi syndrome uniquely deermines wha he error paern was, for single bi errors. So since we know wha he error was, we can correc i and decrease he probabiliy of a received bi error for he sysem! The following able defines he syndrome o error paern relaionship: Page 39

40 Bu if he syndrome is perceived as a binary number, i can be convered o decimal and used as an index ino he following error-paern look-up marix. The decimal value of he syndrome now poins o he row of E ha conains he corresponding error paern. The (7,4) Hamming code is such ha, when here is a single error in y, he row of E chosen wih he syndrome maches he e used above o model he relaionship beween x and y. Therefore, given y, he decoding procedure is y(correced) = y + e (modulo ) = x + e + e (modulo ) = x since addiion modulo of any binary vecor o iself resuls in a vecor of zeroes. Page 40

41 IN Lab Procedure: In his experimen, we will be using Malab o simulae he following sysem: This picure applies o digial communicaion sysems, like compuer modems. Draw a similar block diagram for a CD player. Wha are he sources of noise in such a sysem? We have made available an m-file "coder.m" ha is a skeleon version of he m-file you are o wrie. The crucial saemens, which you mus fill in, are missing. The marices G, H, and E as defined above are se up for you in he leading saemens. In he firs simulaion, you are o proceed as follows in Malab: diary filename1 Avec=; bis=10000; coder(avec,bis) This performs a simulaion for a message sream of lengh bis, i.e. bis/4 messages, each 4 bis in lengh. Avec is a vecor of signal ampliudes A; in he firs simulaion we use only he single value A =. Execuion should ake less han a minue. In he final simulaion we use diary filename Avec=[0.5,1/sqr(),1,sqr(),,*sqr(),4]; bis=00000; coder(avec,bis) which will ake several minues. Therefore in order o avoid wasing a lo of ime you should be sure he firs simulaion is righ by showing he resuls o us before proceeding. In he procedure oulined below, you are asked in several places o record 0 values. We recommend ha his be done only in he firs simulaion wih bis= Wha is asked can be done e.g. wih a simple Malab saemen such as SRC_UNCODED(1:0) Page 41

42 Sep 1 : Generaing he Uncoded Source Sequence The Malab "randn" funcion generaes random numbers wih zero mean and a Gaussian disribuion of uni variance. I can be used in conjuncion wih he "sign" funcion (which reurns +/-1 if a number is greaer han or less han zero, respecively) o generae he sequence of '1's and '0's ha will make up he SRC_UNCODED row vecor. Think of his sequence as a bi sream of sampled music ha will be recorded on a CD or sen o anoher compuer via a modem. => Creae he SRC_UNCODED row vecor of lengh bis. Verify your work by recording he firs 0 bis of SRC_UNCODED. Sep : Generaing he Coded Source Sequence Now we'll generae he coded sequence by applying he (7,4) Hamming code o he SRC_UNCODED sequence. We need o break up he SRC_UNCODED bi sream ino 4-bi sequences ha will be convered ino 7-bi sequences by he coding algorihm. This can be accomplished by a Malab loop ha muliplies 4 bis of SRC_UNCODED (m) by he generaor marix, G, and concaenaes or adds in he resuls (x) o form SRC_CODED. Verify your work by recording he coded versions of he firs 0 bis of SRC_UNCODED ha you wroe down in Sep 1 (his will be more han 0 bis). Do hese agree wih he code able? One migh assume ha, since we are encoding 4-bi sequences (he blocks, m), our original daa sream simulaes a sysem ha produces only nibbles (1 nibble = half of a bye) of daa. Would i make any difference if he daa sream was originally creaed as byes of informaion? as -bye words? Explain Sep 3 : Modulaion A his poin, we are ready o ransmi he binary daa (such as byes on a compuer) over a physical medium (such as an eherne cable). A discree-ime sequence mus be convered ino a coninuous-ime signal for ransmission. This is known as modulaion. We will simulae a modulaion scheme known as BPSK, binary phase-shif keying. The phase-shif here is obviously 180 (alhough BPSK usually refers o shifing he phase of sinusoidal carrier, no DC pulses as we are using here). Page 4

43 In order o accomplish his in our simulaion, we mus conver our SRC sequences of '1's and '0's ino a sequence of '+1' and '-1', and hen muliply by A. => Generae he TX_UNCODED and TX_CODED signals. Verify your work by recording he modulaed levels of he firs 0 pulses in he TX_UNCODED sequence. Sep 4 : Inroducing Noise A his poin, our daa has enered he physical medium and will be modified by addiive whie Gaussian noise. Again, he Malab "randn" funcion will be employed. As menioned in Sep 1, he random numbers produced by his funcion have a "variance" of one. This ells us how much he random numbers vary abou he mean. Thus in he zero mean case, as we have here, he variance is he mean square value of he noise. => Generae a noise sequence equal in lengh o he TX_UNCODED sequence. Plo a hisogram of he noise samples (Malab "his" funcion). A hisogram shows us how many samples occurred over a paricular range of values. Prin ou he hisogram for your lab repor. Wha percenage of he noise samples had values beween -1 and +1? Also verify your work by recording he firs 0 noise samples. => Generae he RX_UNCODED sequence by adding he noise samples o he TX_UNCODED sequence. Verify your work by recording he firs 0 noise samples and he firs 0 RX_UNCODED samples. Compare hem o he TX_UNCODED samples recorded in Sep 3. => Now repea he previous wo operaions o generae RX_CODED by adding noise o each bi of TX_CODED (you will need a longer noise sequence here). For a signal coming from a "random process" such as noise, we are ineresed in is power, raher han energy because he energy of signals ha persis for all ime is infinie. Power, Page 43

44 or energy per uni ime, is he relevan measure here. For zero-mean addiive noise, is power is equal o is variance. For noise, i is also useful o define a posiive-frequency power densiy, N 0. Mahemaically, we admi negaive frequency and consider power as disribued half in negaive frequencies, half in posiive frequencies, and we inegrae he power densiy or specrum over posiive and negaive frequencies o ge he oal power. In all oher respecs we hink of frequency as posiive, since nohing can happen a negaive number of imes per second. Thus N 0, since i relaes o only half he frequencies, mus be wice he power densiy ha we inegrae o ge oal power. So wha is N 0 for our experimen? Anoher issue is he signal power or energy. Here we shall ake wo poins of view. The energy of every bi in he coded case is he same as he energy in he uncoded case, namely, A. Therefore he power in he coded case is he same as for he uncoded case, alhough he former akes 7/4 as long o ransmi. From his poin of view he signal o noise raio (SNR) for he coded case is he same as for he uncoded case, namely, A /N 0. An alernaive viewpoin is ha in he coded case we are using 7/4 imes as much energy, i.e. he energy per message bi is 7/4 wha i is in he uncoded case, so he SNR is (7/4)A /N 0. This is considered here he "adjused" SNR. Sep 5 : Demodulaion A his poin, our simulaed daa has made i hrough he noisy medium and has arrived a is desinaion. The receiver mus now conver he coninuous ime signal back ino a discree one hrough an analog-o-digial conversion called demodulaion or deecion. We will use a simple hreshold deecor referred o as a slicer. Is hreshold is 0V, so any signal above 0 V will be inerpreed as a '1' and any signal below 0 V will be inerpreed as a '0'. => Generae he demodulaed bi sequences DEST_UNCODED and DEST_CODED. Verify your work by recording he firs 0 bis of he DEST_UNCODED row vecor. Compare hese o he 0 original SRC_UNCODED samples from Sep 1 and he noise samples recorded in Sep 4. Did any bi errors occur? Wha were he noise samples ha caused hese receiver errors? Page 44