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Nov '05CS3291 : Section 81 UNIVERSITY of MANCHESTER Department of Computer Science CS3291 Digital Signal Processing 05 Section 8 Introduction to the DFT

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Nov '05CS3291 : Section 82 DTFT of {x[n]} is: If {x[n]} obtained from x a (t), correctly bandlimited then: for - < < : = T DTFT is convenient way of calculating X a (j ) by computer.

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Nov '05CS3291 : Section 83 Two difficulties: (i)Infinite range of summation (ii) X(e j ) is continuous function of Solutions: (i) Time-domain windowing: Restrict {x[n]} to { x[0], x[1], x[N-1]} {x[n]} 0, N-1 (ii) Frequency-domain sampling: Store values of X(e j ) for - < < For real signals we only need 0 < but generalise to complex signals Instead of - < < take 0 < 2

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Nov '05CS3291 : Section 84 Why take 0 < 2 ? X(e j ) = X( e j ( + 2 ) ) for any So for X(e - j / 3 ) look up X(e j 5 / 3 ). Same information, & it is convenient for to start off at 0. In many cases, not interested in >

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Nov '05CS3291 : Section 85 Taking M equally spaced samples over 0 < 2 we get : { X[k] } 0, M-1 { X[0], X[1],, X[M-1] } where X[k] = X(exp(j k )) with k = 2 k/M

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Nov '05CS3291 : Section 86 For spectral analysis, the larger M, the better for drawing accurate graphs etc. But, if we need minimum M for storage of unambiguous spectrum, take M=N. DFT: {x[n]} 0, N-1 {X[k]} 0, N-1 (complex) (complex) k = 2 k/N

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Nov '05CS3291 : Section 87 DFT transforms a finite sequence to another finite sequence. DTFT transforms infinite sequence to continuous functn of Inverse DFT: {X[k]} 0, N-1 {x[n]} 0, N-1 Note Similarity with DFT:

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Nov '05CS3291 : Section 88 Programming the DFT & its inverse: k = 2 k/N Similarity exploited by programs able to perform DFT or its inverse using same code. Programs to implement these equations in a direct manner given in MATLAB (using complex arith) & C (using real arith only). These direct programs are very slow & FFT is much faster.

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Nov '05CS3291 : Section 89 % Given N complex time-domain samples in array x[1:N] E = 2*pi/N ; for k=0 : N-1 X(1+k) = 0 + j*0 ; Wk =k*E ; for L = 0 : N-1 C = cos(L*Wk) + j *sin(L*Wk); X(1+k) = X(1+k) + x(1+L) * C; end; % Now have N complex freq-dom samples in array X[1:N] Direct DFT using complex arithmetic in MATLAB

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Nov '05CS3291 : Section 810 % Given N complex freq-domain samples in array x[1:N] E = -2*pi/N ; for k=0 : N-1 X(1+k) = 0 + j*0 ; Wk =k*E ; for L = 0 : N-1 C = cos(L*Wk) + j *sin(L*Wk); X(1+k) = X(1+k) + x(1+L) * C; end; X(1+k) = X(1+k)/N ; end; % Now have N complex time-dom samples in array X[1:N] Direct inverse DFT using complex arithmetic in MATLAB

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Nov '05CS3291 : Section 811 % Given N complex samples in array x[1:N] if (Invers == 1) E = -2*pi/N else E = 2*pi/N ; for k=0 : N-1 X(1+k) = 0 + j*0 ; Wk =k*E ; for L = 0 : N-1 C = cos(L*Wk) + j *sin(L*Wk); X(1+k) = X(1+k) + x(1+L) * C; end; if (Inverse == 1) X(1+k) = X(1+k)/N ; end; % Now have N complex samples in array X[1:N] Direct forward/inverse DFT using complex arith in MATLAB

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Nov '05CS3291 : Section 812 // Direct fwd/inverse DFT using real arith only in C void directdft(void) // DFT or Inverse DFT by direct method. { // Order=N, Real & imag parts of input in arrays xr & xi // Output:- Real part in array X, Imag part in Y // Invers is 0 for DFT, 1 for IDFT int k, L; float c,e,s,wk; if(Invers==1) e = -2.0*PI/(float)N; else e = 2.0*PI/(float)N; for(k=0;k