A Course Material on PRINCIPLES OF DIGITAL SIGNAL PROCESSING By Mr. A.MUTHUKUMAR ASSISTANT PROFESSOR DEPARTMENT OF ELECTRONICS AND COMMUNICATION ENGINEERING SASURIE COLLEGE OF ENGINEERING VIJAYAMANGALAM – 638 056 QUALITY CERTIFICATE This is to certify that the e‐course material Subject Code : EC6501 Subject : PRINCIPLES OF DIGITAL SIGNAL PROCESSING Class : III Year ECE being prepared by me and it meets the knowledge requirement of the university curriculum. Signature of the Author Name: MUTHUKUMAR.A Designation: AP This is to certify that the course material being prepared by Mr.M.SHANMUGHARAJ is of adequate quality. He has referred more than five books among them minimum one is from abroad author. Signature of HD Name: Dr.K.PANDIARAJAN SEAL EC6502 DIGITAL SIGNAL PROCESSING UNIT I DISCRETE FOURIER TRANSFORM 9 DFT and its properties, Relation between DTFT and DFT, FFT computations using Decimation in time and Decimation in frequency algorithms, Overlap-add and save methods UNIT II INFINITE IMPULSE RESPONSE DIGITAL FILTERS 9 Review of design of analogue Butterworth and Chebyshev Filters, Frequency transformation in analogue domain – Design of IIR digital filters using impulse invariance technique – Design of digital filters using bilinear transform – pre warping – Realization using direct, cascade and parallel forms. UNIT III FINITE IMPULSE RESPONSE DIGITAL FILTERS 9 Symmetric and Antisymmetric FIR filters – Linear phase FIR filters – Design using Hamming, Hanning and Blackmann Windows – Frequency sampling method – Realization of FIR filters – Transversal, Linear phase and Polyphase structures. UNIT IV FINITE WORD LENGTH EFFECTS 9 Fixed point and floating point number representations – Comparison – Truncation and Rounding errors - Quantization noise – derivation for quantization noise power – coefficient quantization error – Product quantization error - Overflow error – Roundoff noise power - limit cycle oscillations due to product roundoff and overflow errors - signal scaling UNIT V MULTIRATE SIGNAL PROCESSING 9 Introduction to Multirate signal processing-Decimation-Interpolation-Polyphase implementation of FIR filters for interpolator and decimator -Multistage implementation of sampling rate conversion- Design of narrow band filters - Applications of Multirate signal processing. TOTAL= 60 PERIODS TEXT BOOKS 1. John G Proakis and Manolakis, “ Digital Signal Processing Principles, Algorithms and Applications”, Pearson, Fourth Edition, 2007. 2. S.Salivahanan, A. Vallavaraj, C. Gnanapriya, Digital Signal Processing, TMH/McGraw Hill International, 2007 REFERENCES 1. E.C.Ifeachor and B.W.Jervis,“Digital signal processing–A practical approach ” Second edition, Pearson, 2002. 2. S.K. Mitra, Digital Signal Processing, A Computer Based approach, Tata Mc GrawHill, 1998. 3. P.P.Vaidyanathan, Multirate Systems & Filter Banks, Prentice Hall, Englewood cliffs, NJ, 1993. 4. Johny R. Johnson, Introduction to Digital Signal Processing, PHI, 2006. S.No Topic Page No. UNIT I - FREQUENCY TRANSFORMATIONS 1 1.1 INTRODUCTION 1 1.2 DIFFERENCE BETWEEN FT & DFT 2 1.3 CALCULATION OF DFT & IDFT 2 1.4 DIFFERENCE BETWEEN DFT & IDFT 3 1.5 PROPERTIES OF DFT 4 1.6 APPLICATION OF DFT 10 1.7 FAST FOURIER ALGORITHM (FFT) 13 1.8 RADIX-2 FFT ALGORITHMS 14 1.9 COMPUTATIONAL COMPLEXITY FFT V/S DIRECT 18 COMPUTATION 1.10 BIT REVERSAL 19 1.11 DECIMATION IN FREQUENCY (DIFFFT) 20 1.12 GOERTZEL ALGORITHM 26 UNIT II - IIR FILTER DESIGN 29 2.1 INTRODUCTION 29 2.2 TYPES OF DIGITAL FILTER 33 2.3 STRUCTURES FOR FIR SYSTEMS 34 2.4 STRUCTURES FOR IIR SYSTEMS 36 2.5 CONVERSION OF ANALOG FILTER INTO DIGITAL FILTER 43 2.6 IIR FILTER DESIGN - BILINEAR TRANSFORMATION 44 METHOD (BZT) 2.7 LPF AND HPF ANALOG BUTTERWORTH FILTER TRANSFER 47 FUNCTION 2.8 METHOD FOR DESIGNING DIGITAL FILTERS USING BZT 47 2.9 BUTTERWORTH FILTER APPROXIMATION 48 2.10 FREQUENCY RESPONSE CHARACTERISTIC 48 2.11 FREQUENCY TRANSFORMATION 48 2.11.1 FREQUENCY TRANSFORMATION (ANALOG FILTER) 48 2.11.2 FREQUENCY TRANSFORMATION (DIGITAL FILTER) 48 UNIT III - FIR FILTER DESIGN 50 3.1 FEATURES OF FIR FILTER 50 3.2 SYMMETRIC AND ANTI-SYMMETRIC FIR FILTERS 50 3.3 GIBBS PHENOMENON 51 3.4 DESIGNING FILTER FROM POLE ZERO PLACEMENT 54 3.5 NOTCH AND COMB FILTERS 54 3.6 DIGITAL RESONATOR 55 UNIT IV – FINITE WORD LENGTH EFFECTS 80 4.1 NUMBER REPRESENTATION 80 4.1.1 FIXED-POINT QUANTIZATION ERRORS 80 4.1.2 FLOATING-POINT QUANTIZATION ERRORS 82 4.2 ROUNDOFF NOISE 83 4.3 ROUNDOFF NOISE IN FIR FILTERS 83 4.4 ROUNDOFF NOISE IN FIXED POINT FIR FILTERS 85 4.5 LIMIT CYCLE OSCILLATIONS 89 4.6 OVERFLOW OSCILLATIONS 90 4.7 COEFFICIENT QUANTIZATION ERROR 91 4.8 REALIZATION CONSIDERATIONS 93 UNIT V - APPLICATIONS OF DSP 95 5.1 SPEECH RECOGNITION 95 5.2 LINEAR PREDICTION OF SPEECH SYNTHESIS 95 5.3 SOUND PROCESSING 97 5.4 ECHO CANCELLATION 97 5.5 VIBRATION ANALYSIS 98 5.6 MULTISTAGE IMPLEMENTATION OF DIGITAL FILTERS 100 5.7 SPEECH SIGNAL PROCESSING 100 5.8 SUBBAND CODING 102 5.9 ADAPTIVE FILTER 107 5.10 AUDIO PROCESSING 110 5.10.1 MUSIC SOUND PROCESSING 110 5.10.2 SPEECH GENERATION 110 5.10.3 SPEECH RECOGNITION 111 5.11 IMAGE ENHANCEMENT 111 UNIT I FREQUENCY TRANSFORMATIONS PRE REQUISITE DISCUSSION: The Discrete Fourier transform is performed for Finite length sequence whereas DTFT is used to perform transformation on both finite and also Infinite length sequence. The DFT and the DTFT can be viewed as the logical result of applying the standard continuous Fourier transform to discrete data. From that perspective, we have the satisfying result that it's not the transform that varies. 1.1 INTRODUCTION Any signal can be decomposed in terms of sinusoidal (or complex exponential) components. Thus the analysis of signals can be done by transforming time domain signals into frequency domain and vice-versa. This transformation between time and frequency domain is performed with the help of Fourier Transform(FT) But still it is not convenient for computation by DSP processors hence Discrete Fourier Transform(DFT) is used. Time domain analysis provides some information like amplitude at sampling instant but does not convey frequency content & power, energy spectrum hence frequency domain analysis is used. For Discrete time signals x(n) , Fourier Transform is denoted as x(ω) & given by ∞ X(ω) = ∑ x (n) e –jωn FT….……(1) n=-∞ DFT is denoted by x(k) and given by (ω= 2 ∏ k/N) N-1 X(k) = ∑ x (n) e –j2 ∏ kn / N DFT…….(2) n=0 IDFT is given as N-1 1 x(n) =1/N ∑ X (k) e j2 ∏ kn / N IDFT……(3) k=0 1.2 DIFFERENCE BETWEEN FT & DFT Sr Fourier Transform (FT) Discrete Fourier Transform (DFT) No 1 FT x(ω) is the continuous DFT x(k) is calculated only at discrete values function of x(n). of ω. Thus DFT is discrete in nature. 2 The range of ω is from - ∏ to ∏ Sampling is done at N equally spaced points over or 0 to 2∏. period 0 to 2∏. Thus DFT is sampled version of FT. 3 FT is given by equation (1) DFT is given by equation (2) 4 FT equations are applicable to most DFT equations are applicable to causal, finite of infinite sequences. duration sequences 5 In DSP processors & computers In DSP processors and computers DFTs are applications of FT are limited mostly used. because x(ω) is continuous APPLICATION function of ω. a) Spectrum Analysis b) Filter Design Q) Prove that FT x(ω) is periodic with period 2∏. Q) Determine FT of x(n)= an u(n) for -1< a < 1. Q) Determine FT of x(n)= A for 0 ≤ n ≤ L-1. Q) Determine FT of x(n)= u(n) Q) Determine FT of x(n)= δ(n) Q) Determine FT of x(n)= e–at u(t) 1.3 CALCULATION OF DFT & IDFT For calculation of DFT & IDFT two different methods can be used. First method is using mathematical – equation & second method is 4 or 8 point DFT. If x(n) is the sequence of N samples then consider WN= e j2 ∏ / N (twiddle factor) Four POINT DFT ( 4-DFT) –j ∏/2 Sr No WN=W4=e Angle Real Imaginary Total 0 1 W4 0 10 1 1 2 W4 - ∏/2 0-j -j 2 3 W4 - ∏ -1 0 -1 3 4 W4 - 3 ∏ /2 0 J J n=0 n=1 n=2 n=3 0 0 0 0 k=0 W4 W4 W4 W4 0 1 2 3 [WN] = k=1 W4 W4 W4 W4 k=2 2 0 2 4 6 W4 W4 W4 W4 0 3 6 9 k=3 W4 W4 W4 W4 Thus 4 point DFT is given as XN = [WN ] XN 1 1 1 1 [WN] = 1 –j -1 j 1 -1 1 -1 1 j -1 -j EIGHT POINT DFT ( 8-DFT) –j Sr No WN = W8= e Angle Magnitude Imaginary Total ∏/4 0 1 W8 0 1----1 1 2 W8 - ∏/4 1/√2-j 1/√21/√2 -j 1/√2 2 3 W8 - ∏/2 0-j -j 3 4 W8 - 3 ∏ /4 -1/√2-j 1/√2- 1/√2-j 1/√2 4 5 W8 - ∏ -1 ---- -1 5 6 W8 - 5∏ / 4 -1/√2+j 1/√2- 1/√2 + j 1/√2 6 7 W8 - 7∏ / 4 0 J J 8 W 7 - 2∏ 1/√2+j 1/√21/√2 + j 1/√2 8 0 8 16 24 32 40 Remember that W8 = W8 = W8 = W8 = W8 = W8 (Periodic Property) Magnitude and phase of x(k) can be obtained as, 2 2 |x(k)| = sqrt ( Xr(k) + XI(k) ) Angle -1 x(k) = tan (XI(k) / XR(k)) Examples: Q) Compute DFT of x(n) = {0,1,2,3} Ans: x4=[6, -2+2j, -2, -2-2j ] Q) Compute DFT of x(n) = {1,0,0,1} Ans: x4=[2, 1+j, 0, 1-j ] Q) Compute DFT of x(n) = {1,0,1,0} Ans: x4=[2, 0, 2, 0 ] Q) Compute IDFT of x(k) = {2, 1+j, 0, 1-j } Ans: x4=[1,0,0,1] 1.4 DIFFERENCE BETWEEN DFT & IDFT Sr DFT (Analysis transform) IDFT (Synthesis transform) No 1 DFT is finite duration discrete IDFT is inverse DFT which is used to calculate time frequency sequence that is obtained domain representation (Discrete time by sampling one period of FT.