Lecture Notes On Analogue Communication Techniques(Module 1 & 2) Topics Covered: 1. Spectral Analysis of Signals 2. Amplitude Modulation Techniques

Transcription

1 Leture Notes On Analogue Communiation Tehniques(Module 1 & ) Topis Covered: 1. Spetral Analysis of Signals. Amplitude Modulation Tehniques 3. Angle Modulation 4. Mathematial Representation of Noise 5. Noise in AM System 6. Noise in FM system

2 Module-I (1 Hours) Spetral Analysis: Fourier Series: The Sampling Funtion, The Response of a linear System, Normalized Power in a Fourier expansion, Impulse Response, Power Spetral Density, Effet of Transfer Funtion on Power Spetral Density, The Fourier Transform, Physial Appreiation of the Fourier Transform, Transform of some useful funtions, Saling, Time-shifting and Frequeny shifting properties, Convolution, Parseval's Theorem, Correlation between waveforms, Auto-and ross orrelation, Expansion in Orthogonal Funtions, Correspondene between signals and Vetors, Distinguishability of Signals. Module-II (14 Hours) Amplitude Modulation Systems: A Method of frequeny translation, Reovery of base band Signal, Amplitude Modulation, Spetrum of AM Signal, The Balaned Modulator, The Square law Demodulator, DSB-SC, SSB-SC and VSB, Their Methods of Generation and Demodulation, Carrier Aquisition, Phase-loked Loop (PLL),Frequeny Division Multiplexing. Frequeny Modulation Systems: Conept of Instantaneous Frequeny, Generalized onept of Angle Modulation, Frequeny modulation, Frequeny Deviation, Spetrum of FM Signal with Sinusoidal Modulation, Bandwidth of FM Signal Narrowband and wideband FM, Bandwidth required for a Gaussian Modulated WBFM Signal, Generation of FM Signal, FM Demodulator, PLL, Pre-emphasis and De-emphasis Filters. Module-III (1 Hours) Mathematial Representation of Noise: Soures and Types of Noise, Frequeny Domain Representation of Noise, Power Spetral Density, Spetral Components of Noise, Response of a Narrow band filter to noise, Effet of a Filter on the Power spetral density of noise, Superposition of Noise, Mixing involving noise, Linear Filtering, Noise Bandwidth, and Quadrature Components of noise. Noise in AM Systems: The AM Reeiver, Super heterodyne Priniple, Calulation of Signal Power and Noise Power in SSB-SC, DSB-SC and DSB, Figure of Merit,Square law Demodulation, The Envelope Demodulation, Threshold Module-IV (8 Hours) Noise in FM System: Mathematial Representation of the operation of the limiter, Disriminator, Calulation of output SNR, omparison of FM and AM, SNR improvement using pre-emphasis, Multiplexing, Threshold infrequeny modulation, The Phase loked Loop. Text Books: 1. Priniples of Communiation Systems by Taub & Shilling, nd Edition. Tata M Graw Hill. Seleted portion from Chapter1, 3, 4, 8, 9 & 10. Communiation Systems by Siman Haykin,4th Edition, John Wiley and Sons In. Referenes Books: 1. Modern digital and analog ommuniation system, by B. P. Lathi, 3rd Edition, Oxford University Press.. Digital and analog ommuniation systems, by L.W.Couh, 6th Edition, Pearson Eduation, Pvt. Ltd.

3 Spetral Analysis of Signals A signal under study in a ommuniation system is generally expressed as a funtion of time or as a funtion of frequeny. When the signal is expressed as a funtion of time, it gives us an idea of how that instantaneous amplitude of the signal is varying with respet to time. Whereas when the same signal is expressed as funtion of frequeny, it gives us an insight of what are the ontributions of different frequenies that ompose up that partiular signal. Basially a signal an be expressed both in time domain and the frequeny domain. There are various mathematial tools that aid us to get the frequeny domain expression of a signal from the time domain expression and vie-versa. FourierSeries is used when the signal in study is a periodi one, whereas Fourier Transform may be used for both periodi as well as non-periodi signals. Fourier Series Let the signal x(t) be a periodi signal with period T 0. The Fourier series of a signal an be obtained, if the following onditions known as the Dirihlet onditions are satisfied: 1. x(t) must be a single valued funtion of t.. x(t) is absolutely integrable over its domain, i.e. x(t) dt = 0 3. The number of maxima and minima of x(t) must be finite in its domain. 4. The number of disontinuities of x(t) must be finite in its domain. A periodi funtion of time, say x(t) having a fundamental period T 0 an be represented as an infinite sum of sinusoidal waveforms, the summation being alled as the Fourier series expansion of the signal. π nt π nt x(t) = A 0 + An os + Bn sin n= 1 T0 n= 1 T0 Where A 0 is the average value of v(t) given by: A 0 T 0 / = 1 T 0 T 0 / x(t)dt And the oeffiients A n and B n are given by A n T0 / π nt = x(t)os dt T T 0 T0 / 0 B n T0 / π nt = x(t)sin dt T T 0 T0 / 0

4 Alternate form of Fourier Series is π nt x(t) = C0 + Cnos φn n= 1 T0 C = A 0 0 C = A + B n n n φ = tan n 1 B A n n The Fourier series hene expresses a periodi signal as an infinite summation of harmonis of fundamental frequeny f 0 =. The oeffiients C T n 0 1 are alled spetral amplitudes i.e. C n is the amplitude of the spetral omponent π nt C n os φn at frequeny nf 0. This form gives one sided T0 spetral representation of a signal as shown in1 st plot of Figure 1. Exponential Form of Fourier Series This form of Fourier series expansion an be expressed as : x(t) = n= Ve j π nt/ T0 T0 1 j π nt/ T0 Vn = x(t) e dt T 0 T0 n The spetral oeffiients V n and V -n have the property that they are omplex onjugates of eah other * V =. This form gives two sided spetral representation of a signal as shown in nd plot of Figure- n V n 1. The oeffiients V n an be related to C n as : V V 0 0 n = C Cn = j n e φ The V n s are the spetral amplitude of spetral omponents j ntf0 Ve π. n

5 C n 0 f 0 f 0 3f 0 frequeny V n -3f 0 -f 0 -f 0 0 f 0 f 0 3f 0 frequeny Figure 1 One sided and orresponding two sided spetral amplitude plot The Sampling Funtion The sampling funtion denoted as Sa(x) is defined as: ( ) Sin x Sa( x) = x And a similar funtion Sin(x) is defined as : ( πx) Sin Sin( x) = πx The Sa(x) is symmetrial about x=0, and is maximum at this point Sa(x)=1. It osillates with an amplitude that dereases with inreasing x. It rosses zero at equal intervals on x at every x = ± nπ, where n is an non-zero integer. Figure Plot of Sin(f)

6 Fourier Transform The Fourier transform is the extension of the Fourier series to the general lass of signals (periodi and nonperiodi). Here, as in Fourier series, the signals are expressed in terms of omplex exponentials of various frequenies, but these frequenies are not disrete. Hene, in this ase, the signal has a ontinuous spetrum as opposed to a disrete spetrum. Fourier Transform of a signal x(t) an be expressed as: F[x(t)] = X(f) = x(t)e j π ft dt x(t) X(f) represents a Fourier Transform pair The time-domain signal x(t) an be obtained from its frequeny domain signal X(f) by Fourier inverse defined as: 1 x(t) = F [ X(f) ] = X(f) e j π ft df When frequeny is defined in terms of angular frequeny ω,then Fourier transform relation an be expressed as: F[x(t)] = X( ω) = x(t)e jωt dt and [ ] jωt x(t) = F X( ω) = X( ω) e dω π 1 1 Properties of Fourier Transform Let there be signals x(t) and y(t),with their Fourier transform pairs: x(t) X(f) y(t) Y(f) then, 1. Linearity Property ax (t) + by(t) ax(f) + by(f), where a and b are the onstants. Duality Property X(t) x( f)or X(t) π X( ω) 3. Time Shift Property x jπ ft0 (t t ) e X(f) 0

7 4. Time Saling Property 1 f x(at) X a a 5. Convolution Property: If onvolution operation between two signals is defined as: ( ) ( ) x(t) y(t) = x τ x t τ dτ, then x(t) y(t) X(f) Y(f) 6. Modulation Property e jπ f t 0 x(t) X(f f ) 7. Parseval s Property x(t) y (t)dt = X(f) Y (f) df 0 8. Autoorrelation Property: If the time autoorrelation of signal x(t) is expressed as: Rx () τ = x(t)x(t τ ) dt,then R ( ) (f) x τ X 9. Differentiation Property: d x (t) j π fx (f) dt Response of a linear system The reason what makes Trigonometri Fourier Series expansion so important is the unique harateristi of the sinusoidal waveform that suh a signal always represent a partiular frequeny. When any linear system is exited by a sinusoidal signal, the response also is a sinusoidal signal of same frequeny. In other words, a sinusoidal waveform preserves its wave-shape throughout a linear system. Hene the response-exitation relationship for a linear system an be haraterised by, how the response amplitude is related to the exitation amplitude (amplitude ratio) and how the response phase is related to the exitation phase (phase differene) for a partiular frequeny. Let the input to a linear system be : v t ( ) j nt i, ω n = Vne ω Then the filter output is related to this input by the Transfer Funtion (harateristi of the Linear j ( n ) H ω = H ω e θ ω, suh that the filter output is given as Filter): ( ) ( ) ( ω ) = ( ω ) n n j ( t j ( )) v t V H e ω θ ω n n o, n n n

8 Normalised Power While disussing ommuniation systems, rather than the absolute power we are interested in another quantity alled Normalised Mean Power. It is an average power normalised aross a 1 ohm resistor, averaged over a single time-period for a periodi signal. In general irrespetive of the fat, whether it is a periodi or non-periodi signal, average normalised power of a signal v(t) is expressed as : T P= lim 1 v () t dt T T T Energy of signal For a ontinuous-time signal, the energy of the signal is expressed as: E= x (t) dt A signal is alled an Energy Signal if 0 < E < P = 0 A signal is alled Power Signal if 0 < P < E = Normalised Power of a Fourier Expansion If a periodi signal an be expressed as a Fourier Series expansion as: ( ) ( π ) ( π ) vt = C+ Cos ft+ Cos 4 ft Then, its normalised average power is given by : T P= lim 1 v () t dt T T T Integral of the ross-produt terms beome zero, sine the integral of a produt of orthogonal signals over period is zero. Hene the power expression beomes: C1 C P= C By generalisation, normalised average power expression for entire Fourier Series beomes:

9 Cn P= C n= 1 In terms of trigonometri Fourier oeffiients A n s, B n s, the power expression an be written as: 0 n n n= 1 n= 1 P= A + A + B In terms of omplex exponential Fourier series oeffiients V n s, the power expressions beomes: P= VV n= * n n Energy Spetral Density(ESD) It an be proved that energy E of a signal x(t) is given by : E = x (t) dt = X (f) df Parseval s Theorem for energy signals So, E= ψ (f) df, where ψ (f) = X (f) Energy Spetral Density The above expression says that ψ (f) integrated over all of the frequenies, gives the total energy of the signal. Hene Energy Spetral Density (ESD) quantifies the energy ontribution from every frequeny omponent in the signal, and is a funtion of frequeny. Power Spetral Density(PSD) It an be proved that the average normalised power P of a signal x(t),suh that periodially repeated version of x(t) suh that x τ τ 1 1 P = lim x () t dt = lim Xτ () t dt τ τ τ τ τ τ τ τ τ x(t); < t < (t) = 0; elsewhere x τ (t) is a trunated and is given by : Parseval s Theorem for power signals So, P = S(f) df, where (f) S(f) lim X τ = τ Power Spetral Density τ The above expression says that S(f)integrated over all of the frequenies, gives the total normalised power of the signal. Hene Power Spetral Density (PSD) quantifies the power ontribution from every frequeny omponent in the signal, and is a funtion of frequeny.

10 Expansion in Orthogonal Funtions Let there be a set of funtions g 1 (x),g (x),g 3 (x),...,g n (x), defined over the interval x 1 < x < x and suh that any two funtions of the set have a speial relation: x g i(x)g j(x)dx = 0. x1 The set of funtions showing the above property are said to be an orthogonal set of funtions in the interval x1 < x < x. We an then write a funtion f (x) in the same interval x 1 < x < x, as a linear sum of suh g n (x) s as: f C1g1 C C3 3 C n n (x) = (x) + g (x) + g (x) g (x), where C n s are the numerial oeffiients The numerial value of any oeffiient C n an be found out as: C n = x x1 x f (x)g (x) dx x1 g n n (x)dx In a speial ase when the funtions g n (x) in the set are hosen suh that g (x) dx =1, then suh a set is alled as a set of orthonormal funtions, that is the funtions are orthogonal to eah other and eah one is a normalised funtion too. x x1 n

11 Amplitude Modulation Systems In ommuniation systems, we often need to design and analyse systems in whih many independent message an be transmitted simultaneously through the same physial transmission hannel. It is possible with a tehnique alled frequeny division multiplexing, in whih eah message is translated in frequeny to oupy a different range of spetrum. This involves an auxiliary signal alled arrier whih determines the amount of frequeny translation. It requires modulation, in whih either the amplitude, frequeny or phase of the arrier signal is varied as aording to the instantaneous value of the message signal. The resulting signal then is alled a modulated signal. When the amplitude of the arrier is hanged as aording to the instantaneous value of the message/baseband signal, it results in Amplitude Modulation. The systems implementing suh modulation are alled as Amplitude modulation systems. Frequeny Translation Frequeny translation involves translating the signal from one region in frequeny to another region. A signal band-limited in frequeny lying in the frequenies from f 1 to f, after frequeny translation an be translated to a new range of frequenies from f 1 to f. The information in the original message signal at baseband frequenies an be reovered bak even from the frequeny-translated signal. The advantagesof frequeny translation are as follows: 1. Frequeny Multiplexing: In a ase when there are more than one soures whih produe bandlimited signals that lie in the same frequeny band. Suh signals if transmitted as suh simultaneously through a transmission hannel, they will interfere with eah other and annot be reovered bak at the intended reeiver. But if eah signal is translated in frequeny suh that they enompass different ranges of frequenies, not interfering with other signal spetrums, then eah signal an be separated bak at the reeiver with the use of proper filters. The output of filters then an be suitably proessed to get bak the original message signal.. Pratiability of antenna: In a wireless medium, antennas are used to radiate and to reeive the signals. The antenna operates effetively, only when the dimension of the antenna is of the order of magnitude of the wavelength of the signal onerned. At baseband low frequenies, wavelength is large and so is the dimension of antenna required is impratiable. By frequeny translation, the signal an be shifted in frequeny to higher range of frequenies. Hene the orresponding wavelength is small to the extent that the dimension of antenna required is quite small and pratial. 3. Narrow banding: For a band-limited signal, an antenna dimension suitable for use at one end of the frequeny range may fall too short or too large for use at another end of the frequeny range. This happens when the ratio of the highest to lowest frequeny ontained in the signal is large (wideband signal). This ratio an be redued to lose around one by translating the signal to a higher frequeny range, the resulting signal being alled as a narrow-banded signal. Narrowband signal works effetively well with the same antenna dimension for both the higher end frequeny as well as lower end frequeny of the band-limited signal. 4. Common Proessing: In order to proess different signals oupying different spetral ranges but similar in general harater, it may always be neessary to adjust the frequeny range of operation of the apparatus. But this may be avoided, by keeping the frequeny range of operation of the apparatus onstant, and instead every time the signal of interest beingtranslated down to the operating frequeny range of the apparatus.

12 Amplitude Modulation Types: 1. Double-sideband with arrier (DSB+C). Double-sideband suppressed arrier (DSB-SC) 3. Single-sideband suppressed arrier (SSB-SC) 4. Vestigial sideband (VSB) Double-sideband with arrier (DSB+C) Let there be a sinusoidal arrier signal (t) = ACos(π f t), of frequeny f. With the onept of amplitude modulation, the instantaneous amplitude of the arrier signal will be modulated (hanged) proportionally aording to the instantaneous amplitude of the baseband or modulating signal x(t).. So the expression for the Amplitude Modulated (AM) wave beomes: [ s(t) = A+ x(t) Cos(π f t) = E(t)Cos s(π ft) E (t) = A+ x(t) ] The time envelope varying amplitude E(t) of the AM wave is alled as the envelope of the AM wave. of the AM wave has the same shape as the message signal or baseband signal. The Figure 3 Amplitude modulation time-domain plot Modulation Index (m a ): It is definedd as the measure of extent of amplitude variation about unmodulated maximumm arrier amplitude. It is also alled as depth of modulation, degree of modulation or modulation fator.

13 m a = x (t) max A On the basis of modulation index, AM signal an be from any of these ases: I. m a > 1 : Here the maximum amplitude of baseband signal exeeds maximum arrier x(t) max > A. In this ase, the baseband signal is not preserved in the AM envelope, amplitude, hene baseband signal reovered from the envelope will be distorted. m : Here the maximum amplitude of baseband signal is less than arrier amplitude II. 1 a x(t) max A. The baseband signal is preserved in the AM envelope. Spetrum of Double-sideband with arrier (DSB+C) Let x(t) be a bandlimited baseband signal with maximum frequeny ontent f m. Let this signal modulate a arrier (t) = ACos(π f t).then the expression for AM wave in time-domain is given by: [ ] s(t) = A+ x(t) Cos(π ft) = ACos(π f t) + x(t)cos(π f t) Taking the Fourier transform of the two terms in the above expression will give us the spetrum of the DSB+C AM signal. 1 ACos(πf t) [ δ(f + f ) + δ(f f ) ] 1 x(t)cos(π ft) (f + f ) + X(f f ) [ X ] So, first transform pair points out two impulses at f = ± f, showing the presene of arrier signal in the modulated waveform. Along with that, the seond transform pair shows that the AM signal spetrum ontains the spetrum of original baseband signal shifted in frequeny in both negative and positive diretion by amount f. The portion of AM spetrum lying from f to f + f in positive m frequeny and from f to f f in negative frequeny represent the Upper Sideband(USB). The m portion of AM spetrum lying from f f to m f in positive frequeny and from f + f to m f in negative frequeny represent the Lower Sideband(LSB). Total AM signal spetrum spans a frequeny from f f to m f + f, hene has a bandwidth of m f m. Power Content in AM Wave By the general expression of AM wave: s(t) = ACos( πf t) + x(t)cos( πf t) Hene, total average normalised power of an AM wave omprises of the arrier power orresponding to first term and sideband power orresponding to seond term of the above expression.

14 P = P + P total arrier sideband T / A (π ft) T / 1 Parrier = lim A Cos dt = T T T / 1 1 Psideband = lim x (t) Cos (π ft) dt x (t) T = T T / In the ase of single-tone modulating signal where x(t) V Cos( π f t) A Parrier = Vm Psideband = 4 A V Ptotal = Parrier + Psideband = + 4 m a Ptotal = Parrier 1 + m = : m m Where, m a is the modulation index given as m a V A m =. Net Modulation Index for Multi-tone Modulation: If modulating signal is a multitone signal expressed in the form: x(t) = V Cos(π f t) + V Cos(πf t) + V Cos(πf t) V Cos( πf t) n n Then, P total m1 m m m = Parrier n Where V1 V V Vn m1 =, m =, m3 =,..., mn = A A A A Generation of DSB+C AM by Square Law Modulation Square law diode modulation makes use of non-linear urrent-voltage harateristis of diode. This method is suited for low voltage levels as the urrent-voltage harateristi of diode is highly nonlinear in the low voltage region. So the diode is biased to operate in this non-linear region for this appliation. A DC battery V is onneted aross the diode to get suh a operating point on the harateristi. When the arrier and modulating signal are applied at the input of diode, different frequeny terms appear at the output of the diode. These when applied aross a tuned iruit tuned to arrier frequeny and a narrow bandwidth just to allow the two pass-bands, the output has the arrier and the sidebands only whih is essentially the DSB+C AM signal.

15 Figure 4 Current-voltage harateristi of diode Figure 5 Square Law Diode Modulator The non-linear urrent voltage relationship an be written in general as: i = av + bv In this appliation v = (t) + x(t) So i = a[acos(π f t) + x(t)] + b[acos(π f t) + x(t)] i = i = a ACos(π f t) + ax(t) + ba Cos (π f t) + b x (t) + ba x(t)cos(π f t) ba aacos( π f t) + a x(t) + ba Cos( π (f )t) + Out of the above frequeny terms, only the boxed terms have the frequenies in the passband of the tuned iruit, and hene will be at the output of the tuned iruit. There is arrier frequeny term and the sideband term whih omprise essentially a DSB+C AM signal. + b x(t) + ba x(t)cos(π f t)

16 Demodulation of DSB+C by Square Law Detetor It an be used to detet modulated signals of small magnitude, so that the operating point may be hosen in the non-linear portion of the V-I harateristi of diode. A DC supply voltage is used to get a fixed operating point in the non-linear region of diode harateristis. The output diode urrent is hene given by the non-linear expression: i= av (t) + b v FM FM (t) Where v (t) = [A + x(t)]cos( πf t) FM Figure 6 Square Law Detetor This urrent will have terms at baseband frequenies as well as spetral omponents at higher frequenies. The low pass filter omprised of the RC iruit is designed to have ut-off frequeny as the highest modulating frequeny of the band limited baseband signal. It will allow only the baseband frequeny range, so the output of the filter will be the demodulated baseband signal. Linear Diode Detetor or Envelope Detetor This is essentially just a half-wave retifier whih harges a apaitor to a voltage to the peak voltage of the inoming AM waveform. When the input wave's amplitude inreases, the apaitor voltage is inreased via the retifying diode quikly, due a very small RC time-onstant (negligible R) of the harging path. When the input's amplitude falls, the apaitor voltage is redued by being disharged by a bleed resistor R whih auses a onsiderable RC time onstant in the disharge path making disharge proess a slower one as ompared to harging. The voltage aross C does not fall appreiably during the small period of negative half-yle, and by the time next positive half yle appears. This yle again harges the apaitor C to peak value of arrier voltage and thus this proess repeats on. Hene the output voltage aross apaitor C is a spiky envelope of the AM wave, whih is same as the amplitude variation of the modulating signal.

17 Figure 7 Envelope Detetor Double Sideband Suppressed Carrier(DSB-SC) If the arrier is suppressed and only the sidebands are transmitted, this will be a way to saving transmitter power. This will not affet the information ontent of the AM signal as the arrier omponent of AM signal do not arry any information about the baseband signal variation. A DSB+C AM signal is given by: s + (t) = ACos( π f t) +x(t)cos(π π ft) DSB C So, the expression for DSB-SC where the arrierr has been suppressed an be given as: s (t) = x(t)cos( πf t) DSB SC Therefore, a DSB-SC signal is obtained by simply multiplying modulating signal x(t) with the arrier signal. This is aomplished by a produt modulator or mixer. Differene from the the DSB+C being only the absene of arrier omponent, and sine DSBSC has still both the sidebands, spetral span of this DSBSC wave is still f f to m f + f, hene has a m bandwidth of f m. Generation of DSB-SC Signal Figure 8 Produt Modulator

18 This voltage is input to the bandpass filter entre frequeny f and bandwidth f m. Hene it allows the omponent orresponding to the seond term of the v i, whih is our desiredd DSB-SC signal. Demodulation of DSBSC signal Synhronous Detetion: DSB-SC signal is generated at the transmitter by frequeny up-translating the baseband spetrum by the arrier frequeny f. Hene the original baseband signal an be reovered by frequeny down-translating the reeived modulated signal by the same amount. Reovery an be ahieved by multiplying the reeived signal by synhronous arrier signal and then low-pass filtering. A iruit whih an produe an output whih is the produt of two signals input to it is alled a produt modulator. Suh an output when the inputs are the modulating signals and the arrier signal is a DSBSC signal. One suh produt modulator is a balaned modulator. Balaned modulator: v v 1 = Cos(π f t) + x(t) = Cos(π f t) x(t) For diode D 1, the nonlinear v-i relationship beomes: i = av + bv = a Cos (π f t) + x(t)] + b[coss (π f t) + x(t)] 1 1 Similarly, for diode D, i = av + bv = a Cos (π f t) x(t)] + b[coss (π f t) x( (t)] 1 [ [ Now, v i = v3 v4 = ( i 1 i )R (substituting for i 1 and i ) v = R[ax(t) + bx(t)cos(π f t)] i

19 Let the reeived DSB-SC signal is : r(t) = x(t)cos( πf t) So after arrier multipliation, the resulting signal: e(t) = x(t) Cos(π ft).cos(π ft) e(t) = x(t) Cos (π ft) 1 e(t) = x(t) [ 1+ Cos( π (f ) t) ] 1 1 e(t) = x(t) + x(t) Cos( π (f ) t) Figure 9 Synhronous Detetion of DSBSC The low-pass filter having ut-off frequeny f m will only allow the baseband term 1 x(t), whih is in the pass-band of the filter and is the demodulated signal. Single Sideband Suppressed Carrier (SSB-SC) Modulation The lower and upper sidebands are uniquely related to eah other by virtue of their symmetry about arrier frequeny. If an amplitude and phase spetrum of either of the sidebands is known, the other sideband an be obtained from it. This means as far as the transmission of information is onerned, only one sideband is neessary. So bandwidth an be saved if only one of the sidebands is transmitted, and suh a AM signal even without the arrier is alled as Single Sideband Suppressed Carrier signal. It takes half as muh bandwidth as DSB-SC or DSB+C modulation sheme. For the ase of single-tone baseband signal, the DSB-SC signal will have two sidebands : The lower side-band: Cos ( π(f f )t) = Cos (πf t) Cos (πf t) + Sin (πf t)sin(πf t) m m m And the upper side-band: Cos ( π(f + f )t) = Cos (πf t) Cos (πf t) Sin (πf t)sin(πf t) m m m

20 If any one of these sidebands is transmitted, it will be a SSB-SC waveform: s(t) = Cos(π f t) Cos(πft) ± Sin(πf t)sin(πft) SSB m m Where (+) sign represents for the lower sideband, and (-) sign stands for the upper sideband. The x = Cos π, so let x (t) = Sin( π f t) be the Hilbert Transform modulating signal here is (t) ( f t) of x (t). The Hilbert Transform is obtained by simply giving for SSB-SC signal an be written as: s(t) = x(t) Cos( πf t) ± x(t)sin( πf t) Where SSB h m h π m to a signal. So the expression xh (t) is a signal obtained by shifting the phase of every omponent present in x (t) by Generation of SSB-SC signal Frequeny Disrimination Method: π. Figure 10 Frequeny Disrimination Method of SSB SC Generation The filter method of SSB generation produes double sideband suppressed arrier signals (using a balaned modulator), one of whih is then filtered to leave USB or LSB. It uses two filters that have different passband entre frequenies for USB and LSB respetively. The resultant SSB signal is then mixed (heterodyned) to shift its frequeny higher. Limitations: I. This method an be used with pratial filters only if the baseband signal is restrited at its lower edge due to whih the upper and lower sidebands do not overlap with eah other. Hene it is used for speeh signal ommuniation where lowest spetral omponent is 70 Hz and it may be taken as 300 Hz without affeting the intelligibility of the speeh signal. II. The design of band-pass filter beomes quite diffiult if the arrier frequeny is quite higher than the bandwidth of the baseband signal. Phase-Shift Method:

21 Figure 11 Phase shift method of SSB-SC generationn The phase shifting method of SSB generationn uses a phase shift tehnique that auses one of the side bands to be anelled out. It uses two balaned modulators instead of one. The balaned modulators effetively eliminate the arrier. The arrier osillator is applied diretly to the upper balaned modulator along with the audio modulating signal. Then both the arrier and modulating signal are shifted in phase by 90o and applied to the seond, lower, balaned modulator. The two balaned modulator output are then added together algebraially. The phase shifting ation auses one side band to be anelled out when the two balaned modulator outputs are ombined. Demodulation of SSB-SC Signals: The baseband or modulating signal x(t) an be reovered from the SSB-SC signal by using synhronous detetion tehnique. With the help of synhronous detetion method, the spetrum of an SSB-SC signal entered about, is retranslated to the basedand spetrum whih is entered about. The proess of synhronous detetion involves multipliation of the reeived SSB-SC signal with a loally generated arrier. Inoming SSB-SC Multiplier Low x( (t) Pass Filter (LPF) The output of the multiplier will be or or or or

22 When e d (t)is passed through a low-pass filter, the terms entre at of detetor is only the baseband part i.e. 1 (t) x. Vestigial Sideband Modulation(VSB) ± ω are filtered out and the output SSB modulation is suited for transmission of voie signals due to the energy gap that exists in the frequeny range from zero to few hundred hertz. But when signals like video signals whih ontain signifiant frequeny omponents even at very low frequenies, the USB and LSB tend to meet at the arrier frequeny. In suh a ase one of the sidebands is very diffiult to be isolated with the help of pratial filters. This problem is overome by the Vestigial Sideband Modulation. In this modulation tehnique along with one of the sidebands, a gradual ut of the other sideband is also allowed whih omes due to the use of pratial filter. This ut of the other sideband is alled as the vestige. Bandwidth of VSB signal is given by : BW= ( f + f ) ( f f ) = f + f v m m v Where f m bandwidth of bandlimited message signal fv width of the vestige in frequeny

23 Angle Modulation Angle modulation may be defined as the proess in whih the total phase angle of a arrier wave is varied in aordane with the instantaneous value of the modulating or message signal, while amplitude of the arrier remain unhanged. Let the arrier signal be expressed as: (t) = ACos(π f t + θ ) Where φ = π f t+ θ Total phase angle θ phase offset f arrier frequeny So in-order to modulate the total phase angle aording to the baseband signal, it an be done by either hanging the instantaneous arrier frequeny aording to the modulating signal- the ase of Frequeny Modulation, or by hanging the instantaneous phase offset angle aording to the modulating signal- the ase of Phase Modulation. An angle-modulated signal in general an be written as ut () = ACos( φ ()) t where, φ (t) is the total phase of the signal, and its instantaneous frequeny f (t) is given by 1 d fi t t π dt φ () = () Sine u(t) is a band-pass signal, it an be represented as ( ) ( ) = π + θ( ) ut ACos ft t and, therefore instantaneous frequeny f i beomes : 1 d fi t f t π dt θ () = + () For angle modulation, total phase angle an modulated either by making the instantaneous frequeny or the phase offset to vary linearly with the modulating signal. Let m(t) be the message signal, then in a Phase Modulation system we implement to have ( t) θ k m( t) θ = and with onstant f, we get (t) linearly varying with message signal. + p and in an Frequeny Modulation system letting phase offset θ be a onstant, we implement to have ( ) f+ ( ) f t = k m t, to get (t) linearly varying with message signal i f where k p and k f are phase and frequeny sensitivity onstants. i

24 The maximum phase deviation in a PM system is given by: Δ θ = max kp () m t max And the maximum frequeny deviation in FM is given by: max max f () Δ f = k m t max () Δ ω = πk m t f max Single Tone Frequeny Modulation The general expression for FM signal is st () ACos( ωt kf m(t)dt ) = + So for the single tone ase, wheremessage signal is m( t) = VCos( ω t) kv f Then s(t) = ACos ωt+ Sin ( ω m t ) ωm ( ω f ωm ) s(t) = ACos t+ m Sin( t) m Here m f kv f Δω = = Modulation Index ω ω m m Types of Frequeny Modulation High frequeny deviation =>High Bandwidth=> High modulation index=>wideband FM Small frequeny deviation =>Small Bandwidth=> Small modulation index=>narrowband FM Carson s Rule It provides a rule of thumb to alulate the bandwidth of a single-tone FM signal. ( ) ( ) Bandwidth = Δ f + f = 1+ m f m f m If baseband signal is any arbitrary signal having large number of frequeny omponents, this rule an be modified by replaing m by deviation ratio D. f

25 Bandwidth = 1+ D f Then the bandwidth of FM signal is given as: ( ) max Spetrum of a Single-tone Narrowband FM signal A single-tone FM modulated signal is mathematially given as: s(t) = ACos t+ ( ω msin f ( ωmt) ) s(t) = ACos( ω t)cos( msin( ω t) ) ASin( ω t)sin( m Sin( ω t) ) f m f m Sine for narrowband FM modulation index m f <<1, sowe approximate as: Cos( msin( ω t)) 1 and Sin( msin( ω t) ) msin( ω t) f m f m f m And the expression s(t) beomes: s(t) = ACos( ω t) Am Sin( ω t) Sin( ω t) f Am f s(t) = ACos( ωt) + Cos( ω + m t Cos ω m)t m { ω ) ( ω } The above equation represents the NBFM signal. This representation is similar to an AM signal, exept that the lower sideband frequeny has a negative sign. Spetrum of a Single-tone Wideband FM signal A single-tone FM modulated signal is mathematially given as: s(t) = ACos t+ ( ω msin f ( ωmt) ) s(t) = ACos( ω t)cos( msin( ω t) ) ASin( ω t)sin( m Sin( ω t) ) f m f m The FM signal an be expressed in the omplex envelope form as: s(t) = Re Ae jωt+ jmfsin( ωmt) s(t) = Re Ae * e s(t) = Re s(t)* e jmf Sin( ωmt) jωt jωt Where f m s(t) = Ae jm Sin( ω t), whih is a periodi funtion of period 1 f m. The Fouries series expansion of this periodi funtion an be written as:

26 s(t) = Ce π n n= j nfmt Where C n spetral oeffiients are given by 1 fm jπ nfmt Cn = fm s (t) e dt 1 fm 1 fm jm ( ) e f Sin ω m t j πnfmt n = m 1 fm C Af dt Substituting x = π ft m, the above equation beomes, C n A = π π π e jmf Sin(x) jnx dx As the above expression is in the form of n th order Bessels funtion of first kind : 1 Jn(m f) = e π π jmf Sin(x) jnx dx, π therefore we an write C = AJ (m ) n n f So, j nfmt s(t) = AJ n(m f ) e π n= Hene the FM signal in omplex envelope form an be written as: j( nfmt t) s(t) A*Re Jn(m f ) e π + ω = n= s(t) = A* Jn(m f )Cos( πnfmt+ ωt) n= This is the Fourier series representation of Wideband Single-tone FM signal. Its Fourier Transform an be written as: S(f) = A* Jn(m f ){ δ(f + f + nfm) + δ(f f nfm) } n= The spetrum of Wideband Single-tone FM signal indiates the following: 1. WBFM has infinite number of sidebands at frequenies (f ± nf ). m

27 . Spetral amplitude values depends upon 3. The number of signifiant sidebands depends upon the modulation index m f.for m f << < 1, J (m ) 0 f and J (m ) 1 f are only signifiant, whereas for m f >> 1, many signifiant sidebands exists. Methods of Generating FM wave Diret FM: In this method the arrier frequeny is diretly varied inaordane with the inoming message signal to produe a frequeny modulated signal. Indiret FM: This method was first proposed by Armstrong. In thismethod, the modulating wave is first used to produe a narrow-band FMwave, and frequeny multipliation is next used to inrease thefrequeny deviation to the desired level. Diret Method or Parameter Variation Method J (m ). n In this method, the baseband or modulating signal diretly modulates the arrier. The arrier signal is generated with the help of an osillator iruit. This osillator iruit usess a parallel tuned L-C iruit. Thus the frequeny of osillation of the arrier generation is governed by the expression: f ω = 1 LC The arrier frequeny is made to vary in aordane with the baseband or modulating signal by making either L or C depend upon to the baseband signal. Suh an osillator whose frequeny is ontrolled by a modulating signal voltage is alled as Voltage Controlled Osillator. The frequeny of VCO is varied aording to the modulating signal simply by putting shunt voltage variable apaitor (varator/variap) with its tuned iruit. The varator diode is a semiondutor diode whose juntion apaitane hanges with d bias voltage. The apaitor C is made muh smaller than the varator diode apaitane Cd so that the RF voltage from osillator aross the diode is small as ompared to reversee bias d voltage aross the varator diode. Figure 1 Varator diode method of FM generation(diret Method)

28 C d k = = k v v D () ( ) D vd = Vo + x t 1 ωi = L C C ω = i ( + ) o o d 1 1 L C kv 1 o o + D Drawbaks of diret method of FM generation: 1. Generation of arrier signal is diretly affeted by the modulating signal by diretly ontrolling the tank iruit and thus a stable osillator iruit annot be used. So a high order stability in arrier frequeny annot be ahieved.. The non-linearity of the varator diode produes a frequeny variation due to harmonis of the modulating signal and therefore the FM signal is distorted. Indiret method or Armstrong method of FM generation A very high frequeny stability an be ahieved sine in this ase the rystal osillator may be used as a arrier frequeny generator. In this method, first of all a narrowband FMis generated and then frequeny multipliation is used to ause required inreased frequeny deviation.the narrow band FM wave is then passed through a frequeny multiplier to obtain the wide band FM wave. Frequeny multipliation sales up the arrier frequeny as well as the frequeny deviation. The rystal ontrolled osillator provides good frequeny stability. But this sheme does not provide both the desired frequeny deviation and arrier frequeny at the same time. This problem an be solved by using multiple stages of frequeny multipliers and a mixer stages. Figure 13 Narrow Band FM Generation

29 FM Demodulators In order to be able to demodulate FM, a reeiver must produe a signal whose amplitude varies as aording to the frequeny variations of the inoming signals and it should be insensitive to any amplitude variations in FM signal. Insensitivity to amplitude variations is ahieved by having a high gain IF amplifier. Here the signals are amplified to suh a degree that the amplifier runs into limiting. In this way any amplitude variations are removed. Generally a FM demodulator is omposed of two parts: Disriminator and Envelope Detetor.Disriminator is a frequeny seletive network whih onverts the frequeny variations in an input signal in to proportional amplitude variations. Hene when it is input with an FM signal, it an produe an amplitude modulated signal. But it does not generally alter the frequeny variations whih were there in the input signal. So the output of a disriminator is a both frequeny and amplitude modulated signal. This signal an be fed to the Envelope Detetorpart of FM demodulator to get bak the baseband signal Figure 14 Slope detetor Figure 15 Frequeny response of slope detetor Slope detetor: A very simplest form of FM demodulation is known as slope detetion or demodulation. It onsists of a tuned iruit that is tuned to a frequeny slightly offset from the arrier of

30 the signal.as the frequeny of the signals varies up and down in frequeny aording to its modulation, so the signal moves up and down the slope of the tuned iruit. This auses the amplitude of the signal to vary in line with the frequeny variations. In fat, at this point the signal has both frequeny and amplitude variations.it an be seen from the diagram that hanges in the slope of the filter, reflet into the linearity of the demodulation proess.the linearity is very dependent not only on the filter slope as it falls away, but also the tuning of the reeiver - it is neessary to tune the reeiver frequeny to a point where the filter harateristi is relatively linear. The final stage in the proess is to demodulate the amplitude modulation and this an be ahieved using a simple diode iruit. One of the most obvious disadvantages of this simple approah is the fat that both amplitude and frequeny variations in the inoming signal appear at the output. However, the amplitude variations an be removed by plaing a limiter before the detetor. The input signal is a frequeny modulated signal. It is applied to the tuned transformer (T1, C1, C ombination) whih is offset from the entre arrier frequeny. This onverts the inoming signal from just FM to one that has amplitude modulation superimposed upon the signal. This amplitude signal is applied to a simple diode detetor iruit, D1. Here the diode provides the retifiation, while C3 removes any unwanted high frequeny omponents, and R1 provides a load. PLL FM demodulator / detetor:when used as an FM demodulator, the basi phase loked loop an be used without any hanges. With no modulation applied and the arrier in the entre position of the pass-band the voltage on the tune line to the VCO is set to the mid position. However, if the arrier deviates in frequeny, the loop will try to keep the loop in lok. For this to happen the VCO frequeny must follow the inoming signal, and in turn for this to our the tune line voltage must vary. Monitoring the tune line shows that the variations in voltage orrespond to the modulation applied to the signal. By amplifying the variations in voltage on the tune line it is possible to generate the demodulated signal.the PLL FM demodulator is one of the more widely used forms of FM demodulator or detetor these days. Its suitability for being ombined into an integrated iruit, and the small number of external omponents makes PLL FM demodulation ICs an ideal andidate for many iruits these days. Figure 16 PLL FM Demodulator

31 Module III Soures and types of Noise Type of noises are Thermal Noise Shot Noise Additive Noise Multipliative Noise (fading) Gaussian Noise Spike Noise or Impulse Noise Soure of thermal noise are resistive elements in eletrial and eletroni iruits. Current flowing in ondutors an also be an example. Constant agitation at moleular level in all material, whih prevails all over the universe, is another example. In brief any soure whih provides the urrent is the ause of the thermal energy. Soure of shot noise is the solid state semiondutor devies like diode, triode, tetrode, and pentode tubes. The noise whih are additive in nature are known as additive noise. This orrupts message signal. Fading ours beause of signal or noise available at destination from multiple paths. White noise is basially approximated by Gaussian noise as its probability density funtion is Gaussian. Spike noise is observed in FM reeivers beause of low input SNR. Frequeny Domain Representation Noise Figure 3.1: (a) A sample noise waveform. (b) A periodi waveform is generated by repeating the interval in (a) from T/ to T/ n (t) is a non periodi omplete noise where as n (s) (t) is a sample of it and is a periodi noise as shown in above figure 3.1(b). (3.1) (3.)

32 Power Spetrum of Noise Figure 3.: The power spetrum of the waveform Power spetral density of noise at kδf or kδf frequeny interval an be written as (3.3) Mean Power spetral density Total power in the interval: (3.4) (3.5) Representation of Noise Atual noise n (t) whih is a non periodi signal an be represented as (3.6) Where,, (3.7) (3.8) Now we an write (3.9) (3.10) Total power P T is Spetral Component of Noise (3.11) Spetral omponent of noise at k th instant and within an interval of Δf an be represented as as given below. (3.1a) (3.1b)

33 Corresponding power an be written as (3.13) Taking a time t = t 1, suh that os πkδf = 1, we have P k =, similarly Taking a time t = t, suh that os πkδf = 0, we have P k =, Hene (3.14) It is observed that (3.15) (3.16) Let us take two spetral omponents of noise as given by Considering similar analysis as above, we have (3.17a) (3.17b) (3.18) This above explanation indiates noise n (t) is random, Gaussian, and stationary proess, where,,,, are unorrelated random Gaussian random variables. The probability density funtion (pdf) of k and θ k an be given as / k 0 (3.19) The pdf desribes a Reyliegh distribution, where as pdf desribes a Uniform distribution. (3.0) Narrowband Filter Response to Noise In the following figure 3.3, the filter used is a narrow band filter with transfer funtion H (f) and pass band is B Hz. The noise at the input of the filter is n (t). Figure 3.3: Filter response to narrowband noise

34 The noise n (t) to the filter H (f) is a wideband noise, whereas the noise at the output of the same filter is a narrowband noise Δn (t). The amplitude variation of this Δn (t) is small as it ontains very few harmonis. If we redue the pass-band B of the filter to a very small value then the variation in amplitude of Δn (t) will be small and may be a approximated sinusoidal signal. Effet of Filter to Noise PSD The noise sample at the output of the filter an be designated as. (3.1) (3.) (3.3) (3.4) (3.5) (3.6) Mixing Noise with Sinusoid Noise n k (t) mixed with a sinusoidal signal at f o an be written as (3.7) It is already understood that (3.8) In ase of atual noise Δf tends to zero, kδf beomes f and therefore, we an write (3.9) Let us single out two spetral omponents of noise n (t)

35 and (3.30a) (3.30b) kδf and lδf is hosen in suh a manner that f o = [(k + l)/]δf ; this means f o is in the middle of kδf and lδf. Let say lδf > kδf. Now we an define two differene frequeny omponents as given below. pδf = f o kδf = lδf f o. These differene frequeny omponents are also unorrelated as follows.. We find the differene frequeny omponents as (3.31a) (3.31b) n p1 (t) is the differene omponent due to the mixing of frequenies f o and kδf, while n p (t) is the differene omponent due to the mixing of frequenies f o and lδf. Now we are interested to find the expeted values of the produt of n p1 (t) and n p (t). Similar to the last explanation, we have So power at differene frequenies Thus mixing noise with a sinusoid signal results in a frequeny shifting of the original noise by f o. The variane of this shifted noise is found by adding the variane of eah new noise omponent. This is also appliable to two shifted power spetral density plots. Mixing Noise with Noise (3.3) (3.33) (3.34) (3.35)

36 (3.36) Linear Filtering of Noise Thermal noise and Shot noise have similar power spetral density whih an be approximated as the power spetral density (PSD) of the White noise. This PSD is as shown in figure 3.4. Figure 3.4: Power spetral density of noise Figure 3.5: A filter is plaed before a demodulator to limit the noise power input to the demodulator In order to minimize the noise power that is presented to the demodulator of a reeiving system, a filter is introdued before the demodulator as shown in figure 3.5. The bandwidth B of the filter is made as narrow as possible so as to avoid transmitting any unneessary noise to the demodulator. For example, in an AM system in whih the baseband extends to a frequeny of f M, the bandwidth B = f M. In a wideband FM system the bandwidth is proportional to twie the frequeny deviation. Noise and Low Pass Filter One of the filter most frequently used is the simple RC low-pas filter (LPF). The same RC LPF with a 3 db utoff frequeny f has the transfer funtion T.F. of RC Low Pass Filter: (3.37) If PSD of input noise. The PSD of output noise is (3.38)

37 Noise power at the filter output, N o an be expressed as (3.39) (3.40) Ideal Low Pass Filter: (3.41) (3.4) Noise and Band Pass Filter Figure 3.6: A retangular band pass filter (3.43) Noise and Differentiator Transfer funtion of a differentiator is: is applied at the input (3.44) If the differentiator is followed by a retangular low pass filter having a bandwidth B. Noise power at the output of the LPF is (3.45)

38 Noise and Integrator Transfer funtion of an integrator is: (3.46) (3.47) (3.48) Noise Bandwidth The noise bandwidth (B N ) is defined as the bandwidth of an idealized (retangular) filter whih passes the same noise power as does the real filter. As per the definition we an find B N = (π/)f o, where f o is the frequeny at whih the transfer funtion of the atual filter is entered. Quadrature omponents of Noise It is sometimes more advantageous to represent Narrowband noise entred around f 0 as (3.49) These n (t) and n s (t) are known as quadrature omponent of noise. Figure 3.7: Quadrature omponents of noise Now as per the initial notation (3.50) (3.51) Where, K.Δf = f 0, Hene (3.5)

39 (3.53) (3.54) A. M. Reeiver This reeiver as shown in figure 3.8 is apable of proessing an amplitude modulated arrier and reovering the baseband signal. The modulated RF arrier + noise is reeived by the reeiving antenna and submitted to Radio frequeny (RF) amplifier. After a number of operations as indiated in the same figure 3.8, finally baseband signal with some small noise is obtained at the output of the reeiver. Figure 3.8: A reeiving system for amplitude modulated signal Superheterodyne priniple In early days TRF reeivers were used to detet the baseband signal from modulated RF signal. The performane of suh reeiver varies as the inoming RF frequeny varies. This is beause it uses single onversion tehnique. Later double onversion tehnique (frequeny of inoming RF signal hanges two times) is used by some reeiver as shown in figure 3.8. These are known as superheterodyne reeiver. The main idea behind the design of suh reeiver is that: whatever may be the frequeny of the inoming RF signal, the output after first onversion

40 always produes a fixed frequeny known as intermediate frequeny. Due to this the performane of reeiver remains same for all type of inoming RF signal. Calulation of Signal power and noise power in SSB SC SSB SC: Signal Power Figure 3.9: Output of multiplier is Output of baseband filter an be written as (3.55) (3.56) (3.57) The input signal power is (3.58) The output signal power is (3.59) (3.60)

41 Noise Power Figure 3.10: (3.61) SNR, /4 /4 (3.6) Calulation of Signal power and noise power in DSB SC When a baseband signal of frequeny f M is transmitted over a DSB-SC system, the bandwidth of the arrier filter must be f M rather than f M. Thus, along with signal the input noise in the frequeny range f f M to f + f M will ontribute to the output noise, rather than only in the range of f to f + f M as in SSB ase. DSB SC: Signal Power: (3.63) (3.64) (3.65)