Digital Processing of

Size: px
Start display at page:

Download "Digital Processing of"

Transcription

1 Chapter 4 Digital Processing of Continuous-Time Signals 清大電機系林嘉文 Original PowerPoint slides prepared by S. K. Mitra 4-1-1

2 Digital Processing of Continuous-Time Signals Digital processing of a continuous-time time signal involves the following basic steps: 1. Conversion of the continuous-timetime signal into a discrete-time signal 2. Processing of the discrete-time time signal 3. Conversion of the processed discrete-time signal back into a continuous-time signal Conversion of a continuous-time signal into digital form is carried out by an analog-to-digital g (A/D) ) converter The reverse operation of converting a digital signal into a continuous-time signal is performed by a digital-toanalog (D/A) converter Original PowerPoint slides prepared by S. K. Mitra 4-1-2

3 Digital Processing of Continuous-Time Signals Since the A/D conversion takes a finite amount of time, a sample-and-hold (S/H) circuit is used to ensure that the input analog signal remains constant in amplitude until the conversion is complete to minimize representation error To prevent aliasing, an analog anti-aliasing filter is employed before the S/H circuit To smooth the output signal of the D/A converter, which has a staircase-like waveform, an analog reconstruction filter is used Original PowerPoint slides prepared by S. K. Mitra 4-1-3

4 Digital Processing of Continuous-Time Signals Both the anti-aliasing aliasing filter and the reconstruction filter are analog lowpass filters The most widely used IIR digital filter design method is based on the conversion of an analog lowpass prototype Discrete-time time signals in many applications are generated by sampling continuous-time signals There exists an infinite number of continuous-time signals, which when sampled lead to the same discrete-time signal Under certain conditions, it is possible to relate a unique continuous-time signal to a given discrete-time ti signal If these conditions hold, then it is possible to recover the original continuous-time time signal from its sampled values Original PowerPoint slides prepared by S. K. Mitra 4-1-4

5 Effect of Sampling in the Frequency Domain Let g a (t) be a continuous-time time signal that is sampled uniformly at t = nt, generating the sequence g[n] where g[n] =g g a (nt), < n < with T being the sampling period The reciprocal of T is called the sampling frequency F T, F T = 1/T The frequency-domain representation of g a (t) is given by its continuous-time Fourier transform (CTFT): The frequency-domain representation of g[n] is given by its DTFT: Original PowerPoint slides prepared by S. K. Mitra 4-1-5

6 Effect of Sampling in the Frequency Domain To establish the relation between G a (jω) and G(e jω ), we treat the sampling operation mathematically as a multiplication of g a (t) by a periodic impulse train p(t) p(t) consists of a train of ideal impulses with a period T as shown below The multiplication operation yields an impulse train: g () t = g () t p () t = g ( nt ) δ ( t nt ) p a a n= Original PowerPoint slides prepared by S. K. Mitra 4-1-6

7 Effect of Sampling in the Frequency Domain g p (t) is a continuous-time signal consisting of a train of uniformly spaced impulses with the impulse at t = nt weighted by the sampled value g a (nt) of g a (t) at that instant There are two different forms of G p (jω) One form is given by the weighted sum of the CTFTs of Original PowerPoint slides prepared by S. K. Mitra 4-1-7

8 Effect of Sampling in the Frequency Domain To derive the second form, we make use of the Poisson s s sum formula: where Ω T = 2π/T, and Φ(jΩ) is the CTFT of ϕ(t) For t = 0, the above equation reduces to From the frequency shifting property of the CTFT, the CTFT of g a (t)e jψt is given by G a (j(ω + Ψ)) Substituting ϕ(t) = g a (t)e jψt into the above equation, we arrive at Original PowerPoint slides prepared by S. K. Mitra 4-1-8

9 Effect of Sampling in the Frequency Domain By replacing with Ω in the above equation we arrive at the alternative form of the CTFT G p (jω) of g p (t), given by Therefore, G p p(j (jω) is a periodic function of Ω consisting of a sum of shifted and scaled replicas of G a (jω), shifted by integer multiples of Ω T and scaled by 1/T The term on the RHS of the previous equation for k = 0 is the baseband portion of G p (jω), and each of the remaining terms are the frequency translated t portions of G p (jω) The frequency range is called the baseband or Nyquist band Original PowerPoint slides prepared by S. K. Mitra 4-1-9

10 Effect of Sampling in the Frequency Domain Assume g a (t) is a band-limited signal with a CTFT G a (jω) as shown below The spectrum, P(jΩ) of p(t) having a sampling period T = 2π/ΩT is indicated below Original PowerPoint slides prepared by S. K. Mitra

11 Effect of Sampling in the Frequency Domain Two possible spectra of G p (jω) are shown below aliasing If Ω T 2Ω m, there is no overlap between the shifted replicas of G a (jω) generating G p (jω) If Ω T < 2Ω m, there is an overlap of the spectra of the shifted replicas of G a (jω) generating G p (jω) (Aliasing) Original PowerPoint slides prepared by S. K. Mitra

12 Effect of Sampling in the Frequency Domain If Ω T >2Ω m, g a (t) can be recovered exactly from g p (t) by passing it through an ideal lowpass filter H r (jω) with a gain T and a cutoff frequency Ω c greater than Ω m and less than Ω c Ω m as shown below Original PowerPoint slides prepared by S. K. Mitra

13 Effect of Sampling in the Frequency Domain The spectra of the filter and pertinent signals are as below If Ω T < 2Ω m, due to the overlap of the shifted replicas of G a (jω), the spectrum G a (jω) cannot be separated by filtering to recover G a (jω) because of the distortion caused by a part of the replicas immediately outside the baseband folded back or aliased into the baseband Original PowerPoint slides prepared by S. K. Mitra

14 Sampling Theorem Sampling theorem - Let g a (t) be a band-limited signal with CTFT G a (jω) = 0 for Ω > Ω m. Then g a (t) is uniquely determined by its samples g a (nt), n if where Ω T = 2π/T Ω T 2Ω m The condition Ω T 2Ω m is often referred to as the Nyquist condition The frequency Ω T /2 is usually referred to as the folding frequency Original PowerPoint slides prepared by S. K. Mitra

15 Sampling Theorem Given {g a (nt)}, we can recover exactly g a (t) by generating an impulse train and then passing it through an ideal lowpass filter H r (jω) with a gain T and a cutoff frequency Ω c satisfying Ω m < Ω c < (Ω T Ω m ) The highest frequency Ω m contained in g a (t) is usually called the Nyquist frequency since it determines the minimum sampling frequency Ω T = 2Ω m that must be used to fully recover g a (t) from its sampled version The frequency 2Ω m is called the Nyquist rate Original PowerPoint slides prepared by S. K. Mitra

16 Sampling Theorem Oversampling -The sampling frequency is higher than the Nyquist rate Undersampling - The sampling frequency is lower than the Nyquist rate Critical sampling - The sampling frequency is equal to the Nyquist rate In digital telephony, a 3.4 khz signal bandwidth is acceptable for telephone conversation a sampling rate of 8 khz is used in telecommunication In high-quality analog music signal processing, a bandwidth of 20 khz has been determined to preserve the fidelity a sampling rate of 44.1 khz is used in CD music systems Original PowerPoint slides prepared by S. K. Mitra

17 Sampling Theorem Example - Consider the three sinusoidal signals: g 1 (t) = cos(6πt) g 2 (t) = cos(14πt) g 3 (t) = cos(26πt) Their corresponding CTFTs are: G 1 ( jω) = π[δ(ω 6π) +δ(ω +6π)] G 2 ( jω) = π[δ(ω 14π) + δ(ω + 14π)] G 3 3( jω) ) = π[δ(ω [ ( 26π) + δ(ω ( + 26π)] Original PowerPoint slides prepared by S. K. Mitra

18 Sampling Theorem These continuous-time signals sampled at a rate of T = 0.1 sec, i.e., with a sampling frequency rad/sec The sampling process generates the continuous-time impulse trains g 1p (t), g 2p (t), and g 3p (t) Their corresponding CTFTs are given by aliasing Original PowerPoint slides prepared by S. K. Mitra aliasing

19 Sampling Theorem We now derive the relation between the DTFT of g[n] and the CTFT of g p (t) To this end we compare with and make use of g[n] = g a (nt), < n < Observation: We have or, equivalently, As a result Original PowerPoint slides prepared by S. K. Mitra

20 Sampling Theorem We arrive at the desired result given by The relation derived above can be alternately expressed as From or from it follows that G(e( jω ) is obtained from G p p(j (jω) by applying ppy the mapping Ω = ω/t (note, both G(e jω ) are G p (jω) are periodic) Original PowerPoint slides prepared by S. K. Mitra

21 Recovery of the Analog Signal (1/3) We now derive the expression for the output of the ideal lowpass reconstruction filter H r (jω) as a function of the samples g[n] The impulse response h r (t) of the lowpass reconstruction ction filter is obtained by taking the inverse DTFT of H r (jω) Thus, the impulse response is given by The input to the lowpass filter is the impulse train g p (t) Original PowerPoint slides prepared by S. K. Mitra

22 Recovery of the Analog Signal (2/3) Therefore, the output of the ideal lowpass filter is given by: Substituting h r (t) = sin(ω c t) /(Ω T t/2) in the above and assuming for simplicity Ω c = Ω T /2 = π/t, we get sin [ π ( t nt ) / T ] g () t = g[ n] a n= π ( t nt)/ T The ideal band-limited interpolation process is shown below Amplitude Original PowerPoint slides prepared by S. K. Mitra Time

23 Recovery of the Analog Signal (3/3) It can be shown that when Ω c = Ω T /2 in h r r( (0) =1 and h r r( (nt) = 0 for n 0 As a result, from we observe g ( rt ) = g [ r ] = g ( rt ) a a for all integer values of r in the range < r < The above relation holds whether or not the condition of the sampling theorem is satisfied However, for all values of t only if the sampling frequency Ω T satisfies the condition of the sampling theorem Original PowerPoint slides prepared by S. K. Mitra

24 Implication of the Sampling Process Consider again the three continuous-time signals: g 1 (t) = cos(6πt), g 2 (t) = cos(14πt), and g 3 (t) = cos(26πt) The plot of the CTFT G 1p (jω) of g 1p (t) is shown below It is apparent that we can recover any of its frequency- translated versions cos[(20k ± 6)πt] outside the baseband by passing g 1p (t) through an ideal analog bandpass filter with a passband centered at Ω = (20k ± 6)π For example, to recover the signal cos(34πt), it will be necessary to use a bandpass filter with a frequency response Original PowerPoint slides prepared by S. K. Mitra

25 Implication of the Sampling Process Likewise, we can recover the aliased baseband component cos(6πt) from the sampled version of either g 2p (t) or g 3p (t) by passing it through an ideal lowpass filter with a frequency response: There is no aliasing i distortion ti unless the original i continuous-time signal also contains the component cos(6πt) Similarly, from either g 2p (t) or g 3p (t) we can recover any one of the frequency translated versions, including the parent continuous-time signal g 2 (t) or g 3 (t) as the case may be, by employing suitable filters Original PowerPoint slides prepared by S. K. Mitra

26 Sampling of Bandpass Signals (1/3) The conditions for the unique representation of a continuous-time signal by the discrete-time signal obtained by uniform sampling assumed that the continuous-time signal is bandlimited in the frequency range from dc to some frequency Ω m Suchacontinuous-time continuous time signal is commonly referred to as a lowpass signal There are applications where the continuous-time signal is bandlimited to a higher frequency range Ω L Ω Ω H Such a signal is usually referred to as a bandpass signal To prevent aliasing a bandpass signal can of course be sampled at a rate greater than twice the highest frequency, i.e. by ensuring Ω L 2Ω H Original PowerPoint slides prepared by S. K. Mitra

27 Sampling of Bandpass Signals (2/3) The spectrum of the discrete-time signal obtained by sampling a bandpass signal will have spectral gaps with no signal components present in these gaps Moreover, er if Ω H is very large, the sampling rate also has to be very large which may not be practical in some situations A more practical approach is to use under sampling Let Ω = Ω H Ω L define the bandwidth of the bandpass signal Assume that the highest frequency contained in the signal is an integer multiple of the bandwidth, i.e., Ω H = M( Ω) Choose the sampling frequency Ω T to satisfy the condition Ω T = 2( Ω) = 2Ω H/M which is smaller than 2Ω H, the Nyquist rate Original PowerPoint slides prepared by S. K. Mitra

28 Sampling of Bandpass Signals (3/3) Substitute the above expression for Ω T in This leads to As before, G p (jω) consists of a sum of G a (jω) and replicas of G a (jω) shifted by integer multiples of twice the bandwidth Ω and scaled by 1/T The amount of shift for each value of k ensures that there will be no overlap between all shifted replicas no aliasing Original PowerPoint slides prepared by S. K. Mitra

29 Sampling of Bandpass Signals As shown above, g a (t) can be recovered from g p (t) by passing it through an ideal bandpass filter with a passband given by Ω L Ω Ω H and a gain of T Note: Any of the replicas in the lower frequency bands can be retained by ypassing ggg p (t) through bandpass filters with passbands Ω L k( Ω) Ω Ω H k( Ω), 1 k M 1 Original PowerPoint slides prepared by S. K. Mitra

30 Analog Lowpass Filter Specifications Typical magnitude response H a (jω) of analog lowpass filter: In the passband, defined by 0 Ω Ω p, we require 1 δ p H a (jω) 1+ δ p, Ω Ω p, i.e., H a (jω) approximates unity within an error of ± δ p In the stopband, defined by Ω s Ω <, we require H a (jω) δ s, Ω s Ω <, ie i.e., H a (jω) approximates zero within an error of δ s Original PowerPoint slides prepared by S. K. Mitra

31 Analog Lowpass Filter Specifications Ω p - passband edge frequency Ω s - stopband edge frequency δ p - peak ripple value in the passband δ s - peak ripple value in the stopband Peak passband ripple α p = 20log 10 (1 δ p ) Minimum stopband attenuation α s = 20log 10 (δ s ) Original PowerPoint slides prepared by S. K. Mitra

32 Analog Lowpass Filter Specifications Magnitude specifications may alternately be given in a normalized form as indicated below Here, the maximum value of the magnitude in the passband assumed to be unity - Maximum passband deviation, given by the minimum value of the magnitude in the passband 1/A -Maximum stopband magnitude Original PowerPoint slides prepared by S. K. Mitra

33 Analog Lowpass Filter Design Two additional parameters are defined: 1) Transition ratio k = Ω p /Ω s for a lowpass filter k < 1 2) Discrimination parameter usually k 1 << 1 Original PowerPoint slides prepared by S. K. Mitra

34 Butterworth Approximation (1/5) The magnitude-square response of an N-th order analog lowpass Butterworth filter is given by First 2N 1 derivatives of H a (jω) 2 at Ω = 0 are equal to zero The Butterworth lowpass filter thus is said to have a maximally-flat magnitude at Ω =0 Gain in db is G(Ω) = 10log 10 H a (jω) 2 As G(0) = 0 and G(Ω c ) =10log 10 (0.5) = dB Ω c is called the 3-dB cutoff frequency Original PowerPoint slides prepared by S. K. Mitra

35 Butterworth Approximation (2/5) Typical magnitude responses with Ω c = 1 Two parameters completely characterizing a Butterworth lowpass filter are Ω c and N These are determined from the specified banedges Ω p and Ω s, and minimum passband magnitude, and maximum stopband ripple 1/A Original PowerPoint slides prepared by S. K. Mitra

36 Butterworth Approximation (3/5) Ω c and N are thus determined from Solving the above we get Since order N must be an integer, value obtained is rounded up to the next highest integer N is used next to determine Ω c by satisfying either the stopband edge or the passband edge spec exactly If the stopband edge spec is satisfied, then the passband edge spec is exceeded providing a safety margin Original PowerPoint slides prepared by S. K. Mitra

37 Butterworth Approximation (4/5) Transfer function of an analog Butterworth lowpass filter is given by where Denominator D N (s) is known as the Butterworth polynomial of order N Original PowerPoint slides prepared by S. K. Mitra

38 Butterworth Approximation (5/5) Example- Determine the lowest order of a Butterworth lowpass filter with a 1-dB cutoff frequency at 1 khz and a minimum attenuation of 40 db at 5 khz Now which h yields ε 2 = , and 10log10(1/A10(1/A 2 ) = 40 Which yields A 2 = 10,000 Therefore and Hence We choose N = 4 Original PowerPoint slides prepared by S. K. Mitra

39 Chebyshev Approximation (1/3) The magnitude-square response of an N-th order analog lowpass Type 1 Chebyshev filter is given by where T N (Ω) is the Chebyshev polynomial of order N: Typical magnitude response plots of the analog lowpass Type 1 Chebyshev filter are shown below Original PowerPoint slides prepared by S. K. Mitra

40 Chebyshev Approximation (2/3) If Ω = Ω s at the magnitude is equal to 1/A, then Solving the above we get Order N is chosen as the nearest integer greater than or equal to the above value The magnitude-square response of an N-th order analog lowpass Type 2 Chebyshev (also called inverse Chebyshev) filter is given by where T N (Ω) is the Chebyshev polynomial of order N Original PowerPoint slides prepared by S. K. Mitra

41 Chebyshev Approximation (3/3) Typical response plots of Type 2 Chebyshev lowpass filter: The order N of the Type 2 Chebyshev filter is determined by Example - Determine the lowest order of a Chebyshev h lowpass filter with an 1-dB cut-off frequency at 1 KHz and a maximum attenuation of 40 db at 5 KHz Original PowerPoint slides prepared by S. K. Mitra

42 Elliptic Approximation (1/2) The square-magnitude response of an elliptic lowpass filter is given by where R N (Ω) is a rational function of order N satisfying R N (1/Ω) = 1/R N (Ω), with the roots of its numerator lying in the interval 0 < Ω <1 and the roots of its denominator lying in the interval 1< Ω < For given Ω p p,, ε,, Ω s s,, and A, the filter order can be estimated using where Original PowerPoint slides prepared by S. K. Mitra

43 Elliptic Approximation (2/2) Example - Determine the lowest order of an Elliptic lowpass filter with an 1-dB cut-off frequency at 1 KHz and a maximum attenuation of 40 db at 5 KHz Note k =02and 0.2 1/ k 1 = Substituting these values we get k = , ρ 0 = , and ρ = Hence N = (choosing N = 3) Typical magnitude response plots with Ω p =1 are shown below Original PowerPoint slides prepared by S. K. Mitra

44 Analog Lowpass Filter Design Example - Determine the lowest order of an Elliptic lowpass filter with an 1-dB cut-off frequency at 1 KHz and a maximum attenuation of 40 db at 5 KHz Code fragments used [N, Wn] = ellipord(wp, Ws, Rp, Rs, s ); [b, a] = ellip(n, Rp, Rs, Wn, s ); with Wp = 2*pi*1000; Ws = 2*pi*5000; Rp = 1; Rs = 40; Original PowerPoint slides prepared by S. K. Mitra

45 Design of Analog Highpass, Bandpass, and Bandstop Filters Steps involved in the design process: Step 1 Develop specifications of a prototype analog lowpass filter H LP (s) from specifications of desired analog filter H D (s) using a frequency transformation Step 2 Design the prototype analog lowpass filter Step 3 Determine the transfer function H D (s) of desired filter by applying inverse frequency transformation to H LP (s) Let s denote the Laplace transform variable of prototype analog lowpass filter H LP (s) and denote the Laplace transform variable ŝ of desired analog filter H D (ŝ) Then Original PowerPoint slides prepared by S. K. Mitra

46 Analog Highpass Filter Design (1/2) Spectral Transformation: where Ω p is the passband edge frequency of H LP (s) and is the passband edge frequency of H HP (ŝ) On the imaginary axis the transformation is Original PowerPoint slides prepared by S. K. Mitra

47 Analog Highpass Filter Design (2/2) Example - Determine the lowest order of an Butterworth lowpass filter with the specifications: Choose Ω p = 1, then Analog lowpass filter specifications: Ω p = 1, Ω s = 1, α p = 0.1 db, α s = 40 db, Code fragments used [N, Wn] = buttord(1, 4, 0.1, 40, s ); [B, A] = butter(n, Wn, s ); [num, den] = lp2hp(b, A, 2*pi*4000); Original PowerPoint slides prepared by S. K. Mitra

48 Analog Bandpass Filter Design (1/7) Spectral Transformation: where Ω p is the passband edge frequency of H LP (s), and are the lower and upper passband edge frequencies of desired bandpass filter H BP (ŝ) On the imaginary axis the transformation is Original PowerPoint slides prepared by S. K. Mitra

49 Analog Bandpass Filter Design (2/7) Case 1: To make we can either increase any one of the stopband edges or decrease any one of the passband edges as shown below 1) Decrease to larger passband and shorter leftmost transition band 2) Increase to No change in passband and shorter leftmost transition band Original PowerPoint slides prepared by S. K. Mitra

50 Analog Bandpass Filter Design (3/7) Note the condition can also be satisfied by decreasing which is acceptable as the passband is reduced from the desired value Alternately, the condition can be satisfied by increasing which is not acceptable as the upper stop band is reduced from the desired value Original PowerPoint slides prepared by S. K. Mitra

51 Analog Bandpass Filter Design (4/7) Case 2: To make we can either increase any one of the stopband edges or decrease any one of the passband edges as shown below 1) Increase to larger passband and shorter rightmost transition band 2) Decrease to No change in passband and shorter rightmost transition band Original PowerPoint slides prepared by S. K. Mitra

52 Analog Bandpass Filter Design (5/7) Note the condition can also be satisfied by increasing which is acceptable as the passband is reduced from the desired value Alternately, the condition can be satisfied by decreasing which is not acceptable as the lower stop band is reduced from the desired value Original PowerPoint slides prepared by S. K. Mitra

53 Analog Bandpass Filter Design (6/7) Example - Determine the lowest order of an Elliptic bandpass filter with the specifications: Now and, since we choose We choose Ω p = 1 Hence Analog lowpass filter specifications: Ω p = 1, Ω s = 1.4, α p = 1 db, α s = 22 db Original PowerPoint slides prepared by S. K. Mitra

54 Analog Bandpass Filter Design (7/7) Code fragments used: [N, Wn] = ellipord(1, 1.4, 1, 22, s ); [B, A] = ellip(n, 1, 22, Wn, s ); [num, den] = lp2bp(b, A, 2*pi* , 2*pi*25/7); Gain plot: Frequency KHz Original PowerPoint slides prepared by S. K. Mitra

Digital Processing of Continuous-Time Signals

Digital Processing of Continuous-Time Signals Chapter 4 Digital Processing of Continuous-Time Signals 清大電機系林嘉文 cwlin@ee.nthu.edu.tw 03-5731152 Original PowerPoint slides prepared by S. K. Mitra 4-1-1 Digital Processing of Continuous-Time Signals Digital

More information

Analog Lowpass Filter Specifications

Analog Lowpass Filter Specifications Analog Lowpass Filter Specifications Typical magnitude response analog lowpass filter may be given as indicated below H a ( j of an Copyright 005, S. K. Mitra Analog Lowpass Filter Specifications In the

More information

Multirate Digital Signal Processing

Multirate Digital Signal Processing Multirate Digital Signal Processing Basic Sampling Rate Alteration Devices Up-sampler - Used to increase the sampling rate by an integer factor Down-sampler - Used to increase the sampling rate by an integer

More information

IIR Filter Design Chapter Intended Learning Outcomes: (i) Ability to design analog Butterworth filters

IIR Filter Design Chapter Intended Learning Outcomes: (i) Ability to design analog Butterworth filters IIR Filter Design Chapter Intended Learning Outcomes: (i) Ability to design analog Butterworth filters (ii) Ability to design lowpass IIR filters according to predefined specifications based on analog

More information

Final Exam Solutions June 14, 2006

Final Exam Solutions June 14, 2006 Name or 6-Digit Code: PSU Student ID Number: Final Exam Solutions June 14, 2006 ECE 223: Signals & Systems II Dr. McNames Keep your exam flat during the entire exam. If you have to leave the exam temporarily,

More information

Signals and Systems Lecture 6: Fourier Applications

Signals and Systems Lecture 6: Fourier Applications Signals and Systems Lecture 6: Fourier Applications Farzaneh Abdollahi Department of Electrical Engineering Amirkabir University of Technology Winter 2012 arzaneh Abdollahi Signal and Systems Lecture 6

More information

Signals and Systems Lecture 6: Fourier Applications

Signals and Systems Lecture 6: Fourier Applications Signals and Systems Lecture 6: Fourier Applications Farzaneh Abdollahi Department of Electrical Engineering Amirkabir University of Technology Winter 2012 arzaneh Abdollahi Signal and Systems Lecture 6

More information

8: IIR Filter Transformations

8: IIR Filter Transformations DSP and Digital (5-677) IIR : 8 / Classical continuous-time filters optimize tradeoff: passband ripple v stopband ripple v transition width There are explicit formulae for pole/zero positions. Butterworth:

More information

LECTURER NOTE SMJE3163 DSP

LECTURER NOTE SMJE3163 DSP LECTURER NOTE SMJE363 DSP (04/05-) ------------------------------------------------------------------------- Week3 IIR Filter Design -------------------------------------------------------------------------

More information

Advanced Digital Signal Processing Part 2: Digital Processing of Continuous-Time Signals

Advanced Digital Signal Processing Part 2: Digital Processing of Continuous-Time Signals Advanced Digital Signal Processing Part 2: Digital Processing of Continuous-Time Signals Gerhard Schmidt Christian-Albrechts-Universität zu Kiel Faculty of Engineering Institute of Electrical Engineering

More information

NH 67, Karur Trichy Highways, Puliyur C.F, Karur District DEPARTMENT OF INFORMATION TECHNOLOGY DIGITAL SIGNAL PROCESSING UNIT 3

NH 67, Karur Trichy Highways, Puliyur C.F, Karur District DEPARTMENT OF INFORMATION TECHNOLOGY DIGITAL SIGNAL PROCESSING UNIT 3 NH 67, Karur Trichy Highways, Puliyur C.F, 639 114 Karur District DEPARTMENT OF INFORMATION TECHNOLOGY DIGITAL SIGNAL PROCESSING UNIT 3 IIR FILTER DESIGN Structure of IIR System design of Discrete time

More information

Sampling and Reconstruction of Analog Signals

Sampling and Reconstruction of Analog Signals Sampling and Reconstruction of Analog Signals Chapter Intended Learning Outcomes: (i) Ability to convert an analog signal to a discrete-time sequence via sampling (ii) Ability to construct an analog signal

More information

ELEC-C5230 Digitaalisen signaalinkäsittelyn perusteet

ELEC-C5230 Digitaalisen signaalinkäsittelyn perusteet ELEC-C5230 Digitaalisen signaalinkäsittelyn perusteet Lecture 10: Summary Taneli Riihonen 16.05.2016 Lecture 10 in Course Book Sanjit K. Mitra, Digital Signal Processing: A Computer-Based Approach, 4th

More information

Continuous-Time Analog Filters

Continuous-Time Analog Filters ENGR 4333/5333: Digital Signal Processing Continuous-Time Analog Filters Chapter 2 Dr. Mohamed Bingabr University of Central Oklahoma Outline Frequency Response of an LTIC System Signal Transmission through

More information

(i) Understanding of the characteristics of linear-phase finite impulse response (FIR) filters

(i) Understanding of the characteristics of linear-phase finite impulse response (FIR) filters FIR Filter Design Chapter Intended Learning Outcomes: (i) Understanding of the characteristics of linear-phase finite impulse response (FIR) filters (ii) Ability to design linear-phase FIR filters according

More information

ECE 203 LAB 2 PRACTICAL FILTER DESIGN & IMPLEMENTATION

ECE 203 LAB 2 PRACTICAL FILTER DESIGN & IMPLEMENTATION Version 1. 1 of 7 ECE 03 LAB PRACTICAL FILTER DESIGN & IMPLEMENTATION BEFORE YOU BEGIN PREREQUISITE LABS ECE 01 Labs ECE 0 Advanced MATLAB ECE 03 MATLAB Signals & Systems EXPECTED KNOWLEDGE Understanding

More information

Sampling and Signal Processing

Sampling and Signal Processing Sampling and Signal Processing Sampling Methods Sampling is most commonly done with two devices, the sample-and-hold (S/H) and the analog-to-digital-converter (ADC) The S/H acquires a continuous-time signal

More information

Chapter-2 SAMPLING PROCESS

Chapter-2 SAMPLING PROCESS Chapter-2 SAMPLING PROCESS SAMPLING: A message signal may originate from a digital or analog source. If the message signal is analog in nature, then it has to be converted into digital form before it can

More information

EE 230 Lecture 39. Data Converters. Time and Amplitude Quantization

EE 230 Lecture 39. Data Converters. Time and Amplitude Quantization EE 230 Lecture 39 Data Converters Time and Amplitude Quantization Review from Last Time: Time Quantization How often must a signal be sampled so that enough information about the original signal is available

More information

(i) Understanding of the characteristics of linear-phase finite impulse response (FIR) filters

(i) Understanding of the characteristics of linear-phase finite impulse response (FIR) filters FIR Filter Design Chapter Intended Learning Outcomes: (i) Understanding of the characteristics of linear-phase finite impulse response (FIR) filters (ii) Ability to design linear-phase FIR filters according

More information

DIGITAL FILTERS. !! Finite Impulse Response (FIR) !! Infinite Impulse Response (IIR) !! Background. !! Matlab functions AGC DSP AGC DSP

DIGITAL FILTERS. !! Finite Impulse Response (FIR) !! Infinite Impulse Response (IIR) !! Background. !! Matlab functions AGC DSP AGC DSP DIGITAL FILTERS!! Finite Impulse Response (FIR)!! Infinite Impulse Response (IIR)!! Background!! Matlab functions 1!! Only the magnitude approximation problem!! Four basic types of ideal filters with magnitude

More information

Copyright S. K. Mitra

Copyright S. K. Mitra 1 In many applications, a discrete-time signal x[n] is split into a number of subband signals by means of an analysis filter bank The subband signals are then processed Finally, the processed subband signals

More information

Digital Signal Processing

Digital Signal Processing Digital Signal Processing Lecture 9 Discrete-Time Processing of Continuous-Time Signals Alp Ertürk alp.erturk@kocaeli.edu.tr Analog to Digital Conversion Most real life signals are analog signals These

More information

Module 3 : Sampling and Reconstruction Problem Set 3

Module 3 : Sampling and Reconstruction Problem Set 3 Module 3 : Sampling and Reconstruction Problem Set 3 Problem 1 Shown in figure below is a system in which the sampling signal is an impulse train with alternating sign. The sampling signal p(t), the Fourier

More information

Infinite Impulse Response (IIR) Filter. Ikhwannul Kholis, ST., MT. Universitas 17 Agustus 1945 Jakarta

Infinite Impulse Response (IIR) Filter. Ikhwannul Kholis, ST., MT. Universitas 17 Agustus 1945 Jakarta Infinite Impulse Response (IIR) Filter Ihwannul Kholis, ST., MT. Universitas 17 Agustus 1945 Jaarta The Outline 8.1 State-of-the-art 8.2 Coefficient Calculation Method for IIR Filter 8.2.1 Pole-Zero Placement

More information

ECE503: Digital Filter Design Lecture 9

ECE503: Digital Filter Design Lecture 9 ECE503: Digital Filter Design Lecture 9 D. Richard Brown III WPI 26-March-2012 WPI D. Richard Brown III 26-March-2012 1 / 33 Lecture 9 Topics Within the broad topic of digital filter design, we are going

More information

Multirate DSP, part 1: Upsampling and downsampling

Multirate DSP, part 1: Upsampling and downsampling Multirate DSP, part 1: Upsampling and downsampling Li Tan - April 21, 2008 Order this book today at www.elsevierdirect.com or by calling 1-800-545-2522 and receive an additional 20% discount. Use promotion

More information

Sampling of Continuous-Time Signals. Reference chapter 4 in Oppenheim and Schafer.

Sampling of Continuous-Time Signals. Reference chapter 4 in Oppenheim and Schafer. Sampling of Continuous-Time Signals Reference chapter 4 in Oppenheim and Schafer. Periodic Sampling of Continuous Signals T = sampling period fs = sampling frequency when expressing frequencies in radians

More information

EEO 401 Digital Signal Processing Prof. Mark Fowler

EEO 401 Digital Signal Processing Prof. Mark Fowler EEO 4 Digital Signal Processing Prof. Mark Fowler Note Set #34 IIR Design Characteristics of Common Analog Filters Reading: Sect..3.4 &.3.5 of Proakis & Manolakis /6 Motivation We ve seenthat the Bilinear

More information

PROBLEM SET 6. Note: This version is preliminary in that it does not yet have instructions for uploading the MATLAB problems.

PROBLEM SET 6. Note: This version is preliminary in that it does not yet have instructions for uploading the MATLAB problems. PROBLEM SET 6 Issued: 2/32/19 Due: 3/1/19 Reading: During the past week we discussed change of discrete-time sampling rate, introducing the techniques of decimation and interpolation, which is covered

More information

Chapter 7 Filter Design Techniques. Filter Design Techniques

Chapter 7 Filter Design Techniques. Filter Design Techniques Chapter 7 Filter Design Techniques Page 1 Outline 7.0 Introduction 7.1 Design of Discrete Time IIR Filters 7.2 Design of FIR Filters Page 2 7.0 Introduction Definition of Filter Filter is a system that

More information

Islamic University of Gaza. Faculty of Engineering Electrical Engineering Department Spring-2011

Islamic University of Gaza. Faculty of Engineering Electrical Engineering Department Spring-2011 Islamic University of Gaza Faculty of Engineering Electrical Engineering Department Spring-2011 DSP Laboratory (EELE 4110) Lab#4 Sampling and Quantization OBJECTIVES: When you have completed this assignment,

More information

UNIT-II MYcsvtu Notes agk

UNIT-II   MYcsvtu Notes agk UNIT-II agk UNIT II Infinite Impulse Response Filter design (IIR): Analog & Digital Frequency transformation. Designing by impulse invariance & Bilinear method. Butterworth and Chebyshev Design Method.

More information

(Refer Slide Time: 02:00-04:20) (Refer Slide Time: 04:27 09:06)

(Refer Slide Time: 02:00-04:20) (Refer Slide Time: 04:27 09:06) Digital Signal Processing Prof. S. C. Dutta Roy Department of Electrical Engineering Indian Institute of Technology, Delhi Lecture - 25 Analog Filter Design (Contd.); Transformations This is the 25 th

More information

Laboratory Assignment 5 Amplitude Modulation

Laboratory Assignment 5 Amplitude Modulation Laboratory Assignment 5 Amplitude Modulation PURPOSE In this assignment, you will explore the use of digital computers for the analysis, design, synthesis, and simulation of an amplitude modulation (AM)

More information

Design of infinite impulse response (IIR) bandpass filter structure using particle swarm optimization

Design of infinite impulse response (IIR) bandpass filter structure using particle swarm optimization Standard Scientific Research and Essays Vol1 (1): 1-8, February 13 http://www.standresjournals.org/journals/ssre Research Article Design of infinite impulse response (IIR) bandpass filter structure using

More information

Chapter 2: Digitization of Sound

Chapter 2: Digitization of Sound Chapter 2: Digitization of Sound Acoustics pressure waves are converted to electrical signals by use of a microphone. The output signal from the microphone is an analog signal, i.e., a continuous-valued

More information

Lecture Schedule: Week Date Lecture Title

Lecture Schedule: Week Date Lecture Title http://elec3004.org Sampling & More 2014 School of Information Technology and Electrical Engineering at The University of Queensland Lecture Schedule: Week Date Lecture Title 1 2-Mar Introduction 3-Mar

More information

Sampling, interpolation and decimation issues

Sampling, interpolation and decimation issues S-72.333 Postgraduate Course in Radiocommunications Fall 2000 Sampling, interpolation and decimation issues Jari Koskelo 28.11.2000. Introduction The topics of this presentation are sampling, interpolation

More information

Multirate DSP, part 3: ADC oversampling

Multirate DSP, part 3: ADC oversampling Multirate DSP, part 3: ADC oversampling Li Tan - May 04, 2008 Order this book today at www.elsevierdirect.com or by calling 1-800-545-2522 and receive an additional 20% discount. Use promotion code 92562

More information

Music 270a: Fundamentals of Digital Audio and Discrete-Time Signals

Music 270a: Fundamentals of Digital Audio and Discrete-Time Signals Music 270a: Fundamentals of Digital Audio and Discrete-Time Signals Tamara Smyth, trsmyth@ucsd.edu Department of Music, University of California, San Diego October 3, 2016 1 Continuous vs. Discrete signals

More information

EELE 4310: Digital Signal Processing (DSP)

EELE 4310: Digital Signal Processing (DSP) EELE 4310: Digital Signal Processing (DSP) Chapter # 10 : Digital Filter Design (Part One) Spring, 2012/2013 EELE 4310: Digital Signal Processing (DSP) - Ch.10 Dr. Musbah Shaat 1 / 19 Outline 1 Introduction

More information

Brief Introduction to Signals & Systems. Phani Chavali

Brief Introduction to Signals & Systems. Phani Chavali Brief Introduction to Signals & Systems Phani Chavali Outline Signals & Systems Continuous and discrete time signals Properties of Systems Input- Output relation : Convolution Frequency domain representation

More information

Signals and Systems. Lecture 13 Wednesday 6 th December 2017 DR TANIA STATHAKI

Signals and Systems. Lecture 13 Wednesday 6 th December 2017 DR TANIA STATHAKI Signals and Systems Lecture 13 Wednesday 6 th December 2017 DR TANIA STATHAKI READER (ASSOCIATE PROFFESOR) IN SIGNAL PROCESSING IMPERIAL COLLEGE LONDON Continuous time versus discrete time Continuous time

More information

Digital Filters IIR (& Their Corresponding Analog Filters) Week Date Lecture Title

Digital Filters IIR (& Their Corresponding Analog Filters) Week Date Lecture Title http://elec3004.com Digital Filters IIR (& Their Corresponding Analog Filters) 2017 School of Information Technology and Electrical Engineering at The University of Queensland Lecture Schedule: Week Date

More information

APPENDIX A to VOLUME A1 TIMS FILTER RESPONSES

APPENDIX A to VOLUME A1 TIMS FILTER RESPONSES APPENDIX A to VOLUME A1 TIMS FILTER RESPONSES A2 TABLE OF CONTENTS... 5 Filter Specifications... 7 3 khz LPF (within the HEADPHONE AMPLIFIER)... 8 TUNEABLE LPF... 9 BASEBAND CHANNEL FILTERS - #2 Butterworth

More information

Signal processing preliminaries

Signal processing preliminaries Signal processing preliminaries ISMIR Graduate School, October 4th-9th, 2004 Contents: Digital audio signals Fourier transform Spectrum estimation Filters Signal Proc. 2 1 Digital signals Advantages of

More information

Part B. Simple Digital Filters. 1. Simple FIR Digital Filters

Part B. Simple Digital Filters. 1. Simple FIR Digital Filters Simple Digital Filters Chapter 7B Part B Simple FIR Digital Filters LTI Discrete-Time Systems in the Transform-Domain Simple Digital Filters Simple IIR Digital Filters Comb Filters 3. Simple FIR Digital

More information

Chapter 6 CONTINUOUS-TIME, IMPULSE-MODULATED, AND DISCRETE-TIME SIGNALS. 6.6 Sampling Theorem 6.7 Aliasing 6.8 Interrelations

Chapter 6 CONTINUOUS-TIME, IMPULSE-MODULATED, AND DISCRETE-TIME SIGNALS. 6.6 Sampling Theorem 6.7 Aliasing 6.8 Interrelations Chapter 6 CONTINUOUS-TIME, IMPULSE-MODULATED, AND DISCRETE-TIME SIGNALS 6.6 Sampling Theorem 6.7 Aliasing 6.8 Interrelations Copyright c 2005- Andreas Antoniou Victoria, BC, Canada Email: aantoniou@ieee.org

More information

Experiment 8: Sampling

Experiment 8: Sampling Prepared By: 1 Experiment 8: Sampling Objective The objective of this Lab is to understand concepts and observe the effects of periodically sampling a continuous signal at different sampling rates, changing

More information

Speech, music, images, and video are examples of analog signals. Each of these signals is characterized by its bandwidth, dynamic range, and the

Speech, music, images, and video are examples of analog signals. Each of these signals is characterized by its bandwidth, dynamic range, and the Speech, music, images, and video are examples of analog signals. Each of these signals is characterized by its bandwidth, dynamic range, and the nature of the signal. For instance, in the case of audio

More information

CS3291: Digital Signal Processing

CS3291: Digital Signal Processing CS39 Exam Jan 005 //08 /BMGC University of Manchester Department of Computer Science First Semester Year 3 Examination Paper CS39: Digital Signal Processing Date of Examination: January 005 Answer THREE

More information

Multirate Signal Processing Lecture 7, Sampling Gerald Schuller, TU Ilmenau

Multirate Signal Processing Lecture 7, Sampling Gerald Schuller, TU Ilmenau Multirate Signal Processing Lecture 7, Sampling Gerald Schuller, TU Ilmenau (Also see: Lecture ADSP, Slides 06) In discrete, digital signal we use the normalized frequency, T = / f s =: it is without a

More information

Spectral Transformation On the unit circle we have

Spectral Transformation On the unit circle we have 1 s of Objetive - Transform a given lowpass digital transfer funtion G L ( to another digital transfer funtion G D ( that ould be a lowpass, highpass, bandpass or bandstop filter z has been used to denote

More information

ECE438 - Laboratory 7a: Digital Filter Design (Week 1) By Prof. Charles Bouman and Prof. Mireille Boutin Fall 2015

ECE438 - Laboratory 7a: Digital Filter Design (Week 1) By Prof. Charles Bouman and Prof. Mireille Boutin Fall 2015 Purdue University: ECE438 - Digital Signal Processing with Applications 1 ECE438 - Laboratory 7a: Digital Filter Design (Week 1) By Prof. Charles Bouman and Prof. Mireille Boutin Fall 2015 1 Introduction

More information

APPLIED SIGNAL PROCESSING

APPLIED SIGNAL PROCESSING APPLIED SIGNAL PROCESSING 2004 Chapter 1 Digital filtering In this section digital filters are discussed, with a focus on IIR (Infinite Impulse Response) filters and their applications. The most important

More information

Continuous vs. Discrete signals. Sampling. Analog to Digital Conversion. CMPT 368: Lecture 4 Fundamentals of Digital Audio, Discrete-Time Signals

Continuous vs. Discrete signals. Sampling. Analog to Digital Conversion. CMPT 368: Lecture 4 Fundamentals of Digital Audio, Discrete-Time Signals Continuous vs. Discrete signals CMPT 368: Lecture 4 Fundamentals of Digital Audio, Discrete-Time Signals Tamara Smyth, tamaras@cs.sfu.ca School of Computing Science, Simon Fraser University January 22,

More information

Lecture 7 Frequency Modulation

Lecture 7 Frequency Modulation Lecture 7 Frequency Modulation Fundamentals of Digital Signal Processing Spring, 2012 Wei-Ta Chu 2012/3/15 1 Time-Frequency Spectrum We have seen that a wide range of interesting waveforms can be synthesized

More information

System on a Chip. Prof. Dr. Michael Kraft

System on a Chip. Prof. Dr. Michael Kraft System on a Chip Prof. Dr. Michael Kraft Lecture 5: Data Conversion ADC Background/Theory Examples Background Physical systems are typically analogue To apply digital signal processing, the analogue signal

More information

Moving from continuous- to discrete-time

Moving from continuous- to discrete-time Moving from continuous- to discrete-time Sampling ideas Uniform, periodic sampling rate, e.g. CDs at 44.1KHz First we will need to consider periodic signals in order to appreciate how to interpret discrete-time

More information

Final Exam. EE313 Signals and Systems. Fall 1999, Prof. Brian L. Evans, Unique No

Final Exam. EE313 Signals and Systems. Fall 1999, Prof. Brian L. Evans, Unique No Final Exam EE313 Signals and Systems Fall 1999, Prof. Brian L. Evans, Unique No. 14510 December 11, 1999 The exam is scheduled to last 50 minutes. Open books and open notes. You may refer to your homework

More information

4. Design of Discrete-Time Filters

4. Design of Discrete-Time Filters 4. Design of Discrete-Time Filters 4.1. Introduction (7.0) 4.2. Frame of Design of IIR Filters (7.1) 4.3. Design of IIR Filters by Impulse Invariance (7.1) 4.4. Design of IIR Filters by Bilinear Transformation

More information

3 Analog filters. 3.1 Analog filter characteristics

3 Analog filters. 3.1 Analog filter characteristics Chapter 3, page 1 of 11 3 Analog filters This chapter deals with analog filters and the filter approximations of an ideal filter. The filter approximations that are considered are the classical analog

More information

Chapter 5 THE APPLICATION OF THE Z TRANSFORM. 5.6 Transfer Functions for Digital Filters 5.7 Amplitude and Delay Distortion

Chapter 5 THE APPLICATION OF THE Z TRANSFORM. 5.6 Transfer Functions for Digital Filters 5.7 Amplitude and Delay Distortion Chapter 5 THE APPLICATION OF THE Z TRANSFORM 5.6 Transfer Functions for Digital Filters 5.7 Amplitude and Delay Distortion Copyright c 2005- Andreas Antoniou Victoria, BC, Canada Email: aantoniou@ieee.org

More information

Outline. Discrete time signals. Impulse sampling z-transform Frequency response Stability INF4420. Jørgen Andreas Michaelsen Spring / 37 2 / 37

Outline. Discrete time signals. Impulse sampling z-transform Frequency response Stability INF4420. Jørgen Andreas Michaelsen Spring / 37 2 / 37 INF4420 Discrete time signals Jørgen Andreas Michaelsen Spring 2013 1 / 37 Outline Impulse sampling z-transform Frequency response Stability Spring 2013 Discrete time signals 2 2 / 37 Introduction More

More information

Final Exam Solutions June 7, 2004

Final Exam Solutions June 7, 2004 Name: Final Exam Solutions June 7, 24 ECE 223: Signals & Systems II Dr. McNames Write your name above. Keep your exam flat during the entire exam period. If you have to leave the exam temporarily, close

More information

ELEC3104: Digital Signal Processing Session 1, 2013

ELEC3104: Digital Signal Processing Session 1, 2013 ELEC3104: Digital Signal Processing Session 1, 2013 The University of New South Wales School of Electrical Engineering and Telecommunications LABORATORY 4: DIGITAL FILTERS INTRODUCTION In this laboratory,

More information

Chapter 7 Single-Sideband Modulation (SSB) and Frequency Translation

Chapter 7 Single-Sideband Modulation (SSB) and Frequency Translation Chapter 7 Single-Sideband Modulation (SSB) and Frequency Translation Contents Slide 1 Single-Sideband Modulation Slide 2 SSB by DSBSC-AM and Filtering Slide 3 SSB by DSBSC-AM and Filtering (cont.) Slide

More information

CMPT 318: Lecture 4 Fundamentals of Digital Audio, Discrete-Time Signals

CMPT 318: Lecture 4 Fundamentals of Digital Audio, Discrete-Time Signals CMPT 318: Lecture 4 Fundamentals of Digital Audio, Discrete-Time Signals Tamara Smyth, tamaras@cs.sfu.ca School of Computing Science, Simon Fraser University January 16, 2006 1 Continuous vs. Discrete

More information

NOVEMBER 13, 1996 EE 4773/6773: LECTURE NO. 37 PAGE 1 of 5

NOVEMBER 13, 1996 EE 4773/6773: LECTURE NO. 37 PAGE 1 of 5 NOVEMBER 3, 996 EE 4773/6773: LECTURE NO. 37 PAGE of 5 Characteristics of Commonly Used Analog Filters - Butterworth Butterworth filters are maimally flat in the passband and stopband, giving monotonicity

More information

DSP Laboratory (EELE 4110) Lab#10 Finite Impulse Response (FIR) Filters

DSP Laboratory (EELE 4110) Lab#10 Finite Impulse Response (FIR) Filters Islamic University of Gaza OBJECTIVES: Faculty of Engineering Electrical Engineering Department Spring-2011 DSP Laboratory (EELE 4110) Lab#10 Finite Impulse Response (FIR) Filters To demonstrate the concept

More information

EEM478-DSPHARDWARE. WEEK12:FIR & IIR Filter Design

EEM478-DSPHARDWARE. WEEK12:FIR & IIR Filter Design EEM478-DSPHARDWARE WEEK12:FIR & IIR Filter Design PART-I : Filter Design/Realization Step-1 : define filter specs (pass-band, stop-band, optimization criterion, ) Step-2 : derive optimal transfer function

More information

Digital Signal Processing (Subject Code: 7EC2)

Digital Signal Processing (Subject Code: 7EC2) CIITM, JAIPUR (DEPARTMENT OF ELECTRONICS & COMMUNICATION) Notes Digital Signal Processing (Subject Code: 7EC2) Prepared Class: B. Tech. IV Year, VII Semester Syllabus UNIT 1: SAMPLING - Discrete time processing

More information

Discretization of Continuous Controllers

Discretization of Continuous Controllers Discretization of Continuous Controllers Thao Dang VERIMAG, CNRS (France) Discretization of Continuous Controllers One way to design a computer-controlled control system is to make a continuous-time design

More information

Design of FIR Filters

Design of FIR Filters Design of FIR Filters Elena Punskaya www-sigproc.eng.cam.ac.uk/~op205 Some material adapted from courses by Prof. Simon Godsill, Dr. Arnaud Doucet, Dr. Malcolm Macleod and Prof. Peter Rayner 1 FIR as a

More information

Figure z1, Direct Programming Method ... Numerator Denominator... Vo/Vi = N(1+D1) Vo(1+D ) = ViN Vo = ViN-VoD

Figure z1, Direct Programming Method ... Numerator Denominator... Vo/Vi = N(1+D1) Vo(1+D ) = ViN Vo = ViN-VoD Z Transform Basics Design and analysis of control systems are usually performed in the frequency domain; where the time domain process of convolution is replaced by a simple process of multiplication of

More information

SAMPLING AND RECONSTRUCTING SIGNALS

SAMPLING AND RECONSTRUCTING SIGNALS CHAPTER 3 SAMPLING AND RECONSTRUCTING SIGNALS Many DSP applications begin with analog signals. In order to process these analog signals, the signals must first be sampled and converted to digital signals.

More information

Signals & Signal Processing

Signals & Signal Processing Chapter 1 Signals & Signal Processing 清大電機系林嘉文 cwlin@ee.nthu.edu.tw 03-5731152 #33120 Original PowerPoint slides prepared by S. K. Mitra 1-1-1 Signal & Signal Processing Signal: quantity that carries information

More information

TE 302 DISCRETE SIGNALS AND SYSTEMS. Chapter 1: INTRODUCTION

TE 302 DISCRETE SIGNALS AND SYSTEMS. Chapter 1: INTRODUCTION TE 302 DISCRETE SIGNALS AND SYSTEMS Study on the behavior and processing of information bearing functions as they are currently used in human communication and the systems involved. Chapter 1: INTRODUCTION

More information

Signals & Signal Processing

Signals & Signal Processing Chapter 1 Signals & Signal Processing 清大電機系林嘉文 cwlin@ee.nthu.edu.tw 03-5731152 #33120 Original PowerPoint slides prepared by S. K. Mitra 1-1-1 Signal & Signal Processing Signal: quantity that carries information

More information

GEORGIA INSTITUTE OF TECHNOLOGY. SCHOOL of ELECTRICAL and COMPUTER ENGINEERING. ECE 2026 Summer 2018 Lab #8: Filter Design of FIR Filters

GEORGIA INSTITUTE OF TECHNOLOGY. SCHOOL of ELECTRICAL and COMPUTER ENGINEERING. ECE 2026 Summer 2018 Lab #8: Filter Design of FIR Filters GEORGIA INSTITUTE OF TECHNOLOGY SCHOOL of ELECTRICAL and COMPUTER ENGINEERING ECE 2026 Summer 2018 Lab #8: Filter Design of FIR Filters Date: 19. Jul 2018 Pre-Lab: You should read the Pre-Lab section of

More information

Team proposals are due tomorrow at 6PM Homework 4 is due next thur. Proposal presentations are next mon in 1311EECS.

Team proposals are due tomorrow at 6PM Homework 4 is due next thur. Proposal presentations are next mon in 1311EECS. Lecture 8 Today: Announcements: References: FIR filter design IIR filter design Filter roundoff and overflow sensitivity Team proposals are due tomorrow at 6PM Homework 4 is due next thur. Proposal presentations

More information

!"!#"#$% Lecture 2: Media Creation. Some materials taken from Prof. Yao Wang s slides RECAP

!!##$% Lecture 2: Media Creation. Some materials taken from Prof. Yao Wang s slides RECAP Lecture 2: Media Creation Some materials taken from Prof. Yao Wang s slides RECAP #% A Big Umbrella Content Creation: produce the media, compress it to a format that is portable/ deliverable Distribution:

More information

CHAPTER 8 ANALOG FILTERS

CHAPTER 8 ANALOG FILTERS ANALOG FILTERS CHAPTER 8 ANALOG FILTERS SECTION 8.: INTRODUCTION 8. SECTION 8.2: THE TRANSFER FUNCTION 8.5 THE SPLANE 8.5 F O and Q 8.7 HIGHPASS FILTER 8.8 BANDPASS FILTER 8.9 BANDREJECT (NOTCH) FILTER

More information

F I R Filter (Finite Impulse Response)

F I R Filter (Finite Impulse Response) F I R Filter (Finite Impulse Response) Ir. Dadang Gunawan, Ph.D Electrical Engineering University of Indonesia The Outline 7.1 State-of-the-art 7.2 Type of Linear Phase Filter 7.3 Summary of 4 Types FIR

More information

y(n)= Aa n u(n)+bu(n) b m sin(2πmt)= b 1 sin(2πt)+b 2 sin(4πt)+b 3 sin(6πt)+ m=1 x(t)= x = 2 ( b b b b

y(n)= Aa n u(n)+bu(n) b m sin(2πmt)= b 1 sin(2πt)+b 2 sin(4πt)+b 3 sin(6πt)+ m=1 x(t)= x = 2 ( b b b b Exam 1 February 3, 006 Each subquestion is worth 10 points. 1. Consider a periodic sawtooth waveform x(t) with period T 0 = 1 sec shown below: (c) x(n)= u(n). In this case, show that the output has the

More information

Concordia University. Discrete-Time Signal Processing. Lab Manual (ELEC442) Dr. Wei-Ping Zhu

Concordia University. Discrete-Time Signal Processing. Lab Manual (ELEC442) Dr. Wei-Ping Zhu Concordia University Discrete-Time Signal Processing Lab Manual (ELEC442) Course Instructor: Dr. Wei-Ping Zhu Fall 2012 Lab 1: Linear Constant Coefficient Difference Equations (LCCDE) Objective In this

More information

Final Exam Practice Questions for Music 421, with Solutions

Final Exam Practice Questions for Music 421, with Solutions Final Exam Practice Questions for Music 4, with Solutions Elementary Fourier Relationships. For the window w = [/,,/ ], what is (a) the dc magnitude of the window transform? + (b) the magnitude at half

More information

In The Name of Almighty. Lec. 2: Sampling

In The Name of Almighty. Lec. 2: Sampling In The Name of Almighty Lec. 2: Sampling Lecturer: Hooman Farkhani Department of Electrical Engineering Islamic Azad University of Najafabad Feb. 2016. Email: H_farkhani@yahoo.com A/D and D/A Conversion

More information

Filter Approximation Concepts

Filter Approximation Concepts 6 (ESS) Filter Approximation Concepts How do you translate filter specifications into a mathematical expression which can be synthesized? Approximation Techniques Why an ideal Brick Wall Filter can not

More information

2.1 BASIC CONCEPTS Basic Operations on Signals Time Shifting. Figure 2.2 Time shifting of a signal. Time Reversal.

2.1 BASIC CONCEPTS Basic Operations on Signals Time Shifting. Figure 2.2 Time shifting of a signal. Time Reversal. 1 2.1 BASIC CONCEPTS 2.1.1 Basic Operations on Signals Time Shifting. Figure 2.2 Time shifting of a signal. Time Reversal. 2 Time Scaling. Figure 2.4 Time scaling of a signal. 2.1.2 Classification of Signals

More information

ECE 5650/4650 Exam II November 20, 2018 Name:

ECE 5650/4650 Exam II November 20, 2018 Name: ECE 5650/4650 Exam II November 0, 08 Name: Take-Home Exam Honor Code This being a take-home exam a strict honor code is assumed. Each person is to do his/her own work. Bring any questions you have about

More information

Frequency-Response Masking FIR Filters

Frequency-Response Masking FIR Filters Frequency-Response Masking FIR Filters Georg Holzmann June 14, 2007 With the frequency-response masking technique it is possible to design sharp and linear phase FIR filters. Therefore a model filter and

More information

Lecture 3 Review of Signals and Systems: Part 2. EE4900/EE6720 Digital Communications

Lecture 3 Review of Signals and Systems: Part 2. EE4900/EE6720 Digital Communications EE4900/EE6720: Digital Communications 1 Lecture 3 Review of Signals and Systems: Part 2 Block Diagrams of Communication System Digital Communication System 2 Informatio n (sound, video, text, data, ) Transducer

More information

SAMPLING THEORY. Representing continuous signals with discrete numbers

SAMPLING THEORY. Representing continuous signals with discrete numbers SAMPLING THEORY Representing continuous signals with discrete numbers Roger B. Dannenberg Professor of Computer Science, Art, and Music Carnegie Mellon University ICM Week 3 Copyright 2002-2013 by Roger

More information

SIGNALS AND SYSTEMS LABORATORY 13: Digital Communication

SIGNALS AND SYSTEMS LABORATORY 13: Digital Communication SIGNALS AND SYSTEMS LABORATORY 13: Digital Communication INTRODUCTION Digital Communication refers to the transmission of binary, or digital, information over analog channels. In this laboratory you will

More information

PROBLEM SET 5. Reminder: Quiz 1will be on March 6, during the regular class hour. Details to follow. z = e jω h[n] H(e jω ) H(z) DTFT.

PROBLEM SET 5. Reminder: Quiz 1will be on March 6, during the regular class hour. Details to follow. z = e jω h[n] H(e jω ) H(z) DTFT. PROBLEM SET 5 Issued: 2/4/9 Due: 2/22/9 Reading: During the past week we continued our discussion of the impact of pole/zero locations on frequency response, focusing on allpass systems, minimum and maximum-phase

More information

Digital Filters FIR and IIR Systems

Digital Filters FIR and IIR Systems Digital Filters FIR and IIR Systems ELEC 3004: Systems: Signals & Controls Dr. Surya Singh (Some material adapted from courses by Russ Tedrake and Elena Punskaya) Lecture 16 elec3004@itee.uq.edu.au http://robotics.itee.uq.edu.au/~elec3004/

More information

Signal Processing. Introduction

Signal Processing. Introduction Signal Processing 0 Introduction One of the premiere uses of MATLAB is in the analysis of signal processing and control systems. In this chapter we consider signal processing. The final chapter of the

More information

Filter Banks I. Prof. Dr. Gerald Schuller. Fraunhofer IDMT & Ilmenau University of Technology Ilmenau, Germany. Fraunhofer IDMT

Filter Banks I. Prof. Dr. Gerald Schuller. Fraunhofer IDMT & Ilmenau University of Technology Ilmenau, Germany. Fraunhofer IDMT Filter Banks I Prof. Dr. Gerald Schuller Fraunhofer IDMT & Ilmenau University of Technology Ilmenau, Germany 1 Structure of perceptual Audio Coders Encoder Decoder 2 Filter Banks essential element of most

More information