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UNIVERSITY of MANCHESTER Department of Computer Science CS3291 Digital Signal Processing ‘05

UNIVERSITY of MANCHESTER Department of Computer Science CS3291 Digital Signal Processing ‘05 Section 8 Introduction to the DFT. DTFT of {x[n]} is:. If {x[n]} obtained from x a (t), correctly bandlimited then:. for -  <  <  :  =  T

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UNIVERSITY of MANCHESTER Department of Computer Science CS3291 Digital Signal Processing ‘05

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  1. UNIVERSITY of MANCHESTER Department of Computer Science CS3291 Digital Signal Processing ‘05 Section 8 Introduction to the DFT CS3291 : Section 8

  2. DTFT of {x[n]} is: If {x[n]} obtained from xa(t), correctly bandlimited then: for - <  < :  = T DTFT is convenient way of calculating Xa(j) by computer. CS3291 : Section 8

  3. 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(ej) for - <  <  • For real signals we only need 0  <  but generalise to complex signals • Instead of - <  <  take 0  < 2 CS3291 : Section 8

  4. 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  >  CS3291 : Section 8

  5. 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 CS3291 : Section 8

  6. 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 CS3291 : Section 8

  7. 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: CS3291 : Section 8

  8. 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. CS3291 : Section 8

  9. Direct DFT using complex arithmetic in MATLAB % 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; end; % Now have N complex freq-dom samples in array X[1:N] CS3291 : Section 8

  10. Direct inverse DFT using complex arithmetic in MATLAB % 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] CS3291 : Section 8

  11. Direct forward/inverse DFT using complex arith in MATLAB % 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] CS3291 : Section 8

  12. // 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<N;k++) { x[k]=0.0; y[k]=0.0 ; wk=(float)k*e ; for(L=0; L<N; L++) { c=cos(L*wk); s=sin(L*wk); x[k] = x[k] + xr[L]*c + xi[L]*s; y[k] = y[k] + xi[L]*c - xr[L]*s;} if(Invers==1) { x[k]=x[k]/(float)N; y[k]=y[k]/(float)N;} } } CS3291 : Section 8

  13. Fast Fourier Transform (FFT) • An FFT algorithm, programmed in "C ", is presented in Table 2. • Gives ‘exactly’ same results as DFT only much faster. • Its detail & how speed is achieved is outside our syllabus. • We are interested in how to use DFT & interpret its results. • MATLAB has efficient fft procedure in its ‘SP tool-box'. • Don’t need to know how it’s programmed, only how to use it! • Direct DFT programs of academic interest only. • Table 4 is MATLAB program which reads music from a 'wav' file, splits it up into 512 sample sections & • performs a DFT (by FFT) analysis on each section. CS3291 : Section 8

  14. Effect of windowing (old) • DFT of {x[0], x[1], ..., x[N-1] } is frequency-sampled version of DTFT of infinite sequence {x[n]} with all samples outside range n= 0 to N-1 set to zero. • {x[n]} effectively multiplied by "rectangular window" {r[n]} When N is even, the DTFT, R(ej) of {r[n]} is: Dirichlet Kernel CS3291 : Section 8

  15. Effect of windowing (newer) • Given {x[n]}0,N-1 {x[0], x[1], ..., x[N-1] } • Assumed obtained by multiplying infinite sequence {x[n]} by "rectangular window" {r[n]} When N is even, the DTFT, R(ej) of {r[n]} is: Dirichlet Kernel CS3291 : Section 8

  16. n 0 N • Effect of windowing • Given {x[n]}0,N-1 {x[0], x[1], ..., x[N-1] } • Assumed obtained by multiplying infinite sequence {x[n]} by "rectangular window" {r[n]} r[n] CS3291 : Section 8

  17. When N is even, the DTFT, R(ej) of {r[n]} is: Dirichlet Kernel CS3291 : Section 8

  18. sincs10(/(2)) plotted against : -  CS3291 : Section 8

  19. -  CS3291 : Section 8

  20. Notation “sincsM(/(2))” is not in textbooks. • This function is also encountered when designing FIR filters. • Causes the stop-band ripples which appear when FIR low-pass filters are designed with rectangular windows. • Magnitude of R(ej  ) shown above when M=10. • Note relatively narrow main lobe & side-lobes. • Zero-crossings occur at  = 2 / M , 4 / M, etc. • Like a "sinc" function in many ways. CS3291 : Section 8

  21. Frequency-domain convolution: • DTFT of product of {x[n]} & {r[n]} is complex ( freq.-domain)convolution between X(ej) & R(ej) denoted X(ej)  R(ej): • Observe the form of this expression & its similarity with time-domain convolution formulae. • Also note the (1/2) factor & limits - to  which may remind us of the inverse DTFT. CS3291 : Section 8

  22. Duality of time- & frequency-domain convolution (i) Fourier transform of h(t)  x(t) is H(j).X(j). (ii) DTFT of {h[n]} {x[n]} is H(e j  ).X(e j  ) Time-domain Frequency-domain Convolution (filtering) Multiplication Multiplication (windowing) Complex convolution CS3291 : Section 8

  23. Duality of time- & frequency-domain convolution Time-domain Frequency-domain Convolution (filtering) Multiplication Multiplication (windowing) Complex convolution CS3291 : Section 8

  24. Spectral analysis of sampled sine-waves • DTFT of {cos(0n)} cannot be found directly. • But inverse DTFT of • X(e j  ) = ( - 0) + ( + 0) is: •  • (1/2)((-0) + (+0) ) ejn d •  • = (1/2)[ e j o n + e -j o n ] = cos ( 0 n) for - < n <  • It may be inferred that DTFT of {cos(0n) } is: • ( - 0) + ( + 0) CS3291 : Section 8

  25. X(ej) DTFT of {cos(n)}    -0 - 0  CS3291 : Section 8

  26. X(e j  )    -0 - 0  R(e j( - ) ) • DTFT of rectangularly windowed sampled sine-wave • DTFT of {cos0n}0,N-1 = DTFT of {x[n].r[n]} • = X(ej)  R(ej) = P(ej) say. • By frequency-domain convolution formula,  CS3291 : Section 8

  27.  X(e j )  0 -0 -  R(e j(  ) )  CS3291 : Section 8

  28. X(e j ) = ( - 0) + (  + 0) Now draw this & its modulus: CS3291 : Section 8

  29. P(e j  ) - 0  0 CS3291 : Section 8

  30. | P(e j  ) |  0  0 CS3291 : Section 8

  31. When rectangular window is of width N samples: • Ampl of main peak = ampl of sine-wave*N/2 check! • N-1 zero-crossings between 0 and . • 1 main peak & N-2 ripples in magnitude spectrum. • Increasing N gives sharper peak & more ripples. CS3291 : Section 8

  32. Effect of windowing on DFT : • Effect on DTFT is frequency spreading & side lobes. • How does windowing with frequency sampling affect the DFT? • Effect is rather confusing • Consider 64 pt DFT of { cos(0.7363n)} in Fig 1. • 0 = 0.7363 lies between 7 = (2/64)*7 & 8 = (2/64)*8 • Samples of rectangular window seen. • X[7] and X[8] strongly affected by sinusoid. CS3291 : Section 8

  33. Amplitude  20 Fig 1 : Magnitude of 64 pt DFT spectrum of cos(0.7363n) CS3291 : Section 8

  34. Now consider 64 pt DFT of { cos (n/4) } in Fig 2 • 0 = /4 coincides exactly with 8. • Only X[8] affected. CS3291 : Section 8

  35. Ampl. = 32 Approx. 36% difference in ampl. For same ampl. of sinusoid. Fig2 : Magnitude of 64 point DFT of cos(pn/4). CS3291 : Section 8

  36. What has happened to the ‘sincs’ function in Figure 2 ? • We don’t see it because all the samples apart from one • occur at the ‘zero-crossings of the ‘sincs function’. • Effect of the rectangular window is no longer seen because the sampling points happen to coincide exactly with the zero-crossings of the sincs function. • This always occurs when the frequency of the cosine wave coincides exactly with a frequency sampling point. CS3291 : Section 8

  37. Difference between Figs 1 & 2 undesirable. Use non-rectangular windows e.g. Hann {w[n]}0,N-1 with (Slightly different definition from the one we had for FIR filters). CS3291 : Section 8

  38. CS3291 : Section 8

  39. In frequency-domain : • (i) broader main-lobe for Hann window whose width is approx doubled as compared with rectr window • (ii) reduced side-lobe levels. • For sine-wave, at least 3 spectral 'bins' strongly affected even when its frequency coincides with a sampling point. • Often take highest of the 3 peaks as amplitude of sine-wave. • With rectr : just one bin affected when frequency of sine-wave coincides with a bin. If frequency then shifted towards next bin,  35% variation amplitude seen. • With Hann: Variation of only about 15% seen. • Still undesirable, but not as bad. CS3291 : Section 8

  40. Reduction in amplitude estimation error for sine-waves is at expense of some loss of spectral resolution • With Hann window, 3 bins strongly affected by one sine-wave • We will only know that the frequency of the sine-wave lies within the range of these 3 frequencies. CS3291 : Section 8

  41. DFT & FFT have many applications in DSP esp in comms. • e.g. filtering by (1) performing an FFT, • (2) zeroing unwanted spectral components • (3) performing an inverse FFT. • FFT is ‘Swiss army knife' of signal processing. • Most ‘spectrum analysers’ use an FFT algorithm. • Some applications of spectral estimation are: • determining frequency & loudness of a musical note, • deciding whether a signal is periodic or non-periodic, • analysing speech to recognise spoken words, • looking for sine-waves buried in noise, • measuring frequency distribution of power or energy. CS3291 : Section 8

  42. Spectral analysis of 'power signals' {x[n]} • These exist for all time. • Extract segment {x[n]}0,N-1 to represent {x[n]}. • Energy of {x[n]}0,N-1 is: • Mean square value (MSV) of this segment is (1/N) E • This is “power” of a periodic discrete time signal, of period N samples, for which a single cycle is {x[n]}0,N-1 . • It may be used as an estimate of the power of {x[n]}. CS3291 : Section 8

  43. Parseval's Theorem for the DFT It may be shown that for a real signal segment {x[n]}0,N-1: Proof CS3291 : Section 8

  44. Parseval’s Theorem allows energy & power estimates to be calculated in frequency-domain instead of in time-domain. • Allows us to see how power is distributed in frequency. • Is there more power at high frequencies than at low frequencies or vice-versa? • By Parseval's thm, estimate of power of {x[n]}, obtained by analysing {x[n]}0,N-1 , is: • Usefulness of this estimate illustrated by following example. CS3291 : Section 8

  45. Example: Real periodic signal {x[n]} is rect windowed to give {x[n]}0,39 . 40-point DFT gives magnitude spectrum below. Estimate power of {x[n]} & comment on reliability of estimate. If {x[n]} is passed thro’ ideal digital low-pass filter with cut-off /2 radians/sample how is power likely to be affected? CS3291 : Section 8

  46. Ans: • MSV of {x[n]}0,39 power of {x[n]} • = (1/1600)[2*402 +2*302 +2*202+2*102] = 3.75 watts. • Reduces to 3.125 watts, i.e. by 0.8 dB • Care needed to interpret such results as power estimates. • For periodic or deterministic (non-random) signals: • estimates from segments extracted from different parts of {x[n]} may be similar, & estimates could be fairly reliable. • For random signals : may be considerable variation from estimate to estimate. Averaging may be necessary. CS3291 : Section 8

  47. Power spectral density (PSD) estimate • For N-point DFT, X[k]2 / N2 is estimate of PSD in Watts per “bin”. • A “bin” is a band-width 2/N radians/sample centred on k • ( fS / N Hz centred on k fS / N ) • Instead of |X[k]| often plot 10 log10 (|X[k]|2/N2) dB. against k. • “PSD estimate” graph. • Careful with random signals : each spectral estimate different. • Can average several PSD estimates. CS3291 : Section 8

  48. Spectral analysis of signals: • File: OPERATOR.pcm contains sampled speech. • SNR-12dB.pcm contains sine-wave corrupted with noise. • Sampled at 8 kHz using 12-bit A/D converter. • May be read into "MATLAB" program in Table 5 & spectrally analysed using the FFT. • Meaningless to analyse a large speech file at once. • Divide into blocks of 128 or 256 samples & analyse separately. • Blocks of N (= 512) samples may be read in and displayed. • Programs in notes available on: ~barry/mydocs/CS3291 • In each case, a rectangular window is used. • Exercise, modify programs to have a Hann window. • Notice effectiveness of DFT in locating sine-wave in noise • even when it cannot be seen in time-domain graph. CS3291 : Section 8

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