UNIVERSITY of MANCHESTER Department of Computer Science CS3291 Digital Signal Processing ‘05
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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 - PowerPoint PPT Presentation
Transcript of UNIVERSITY of MANCHESTER Department of Computer Science CS3291 Digital Signal Processing ‘05
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UNIVERSITY of MANCHESTER
Department of Computer Science
CS3291
Digital Signal Processing 05
Section 8
Introduction to the DFT
CS3291 : Section 8
- DTFT of {x[n]} is:If {x[n]} obtained from xa(t), correctly bandlimited then: for - <
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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
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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
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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(jk)) with k = 2k/M
CS3291 : Section 8
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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 = 2k/N
CS3291 : Section 8
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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-1Note Similarity with DFT:
CS3291 : Section 8
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Programming the DFT & its inverse:k = 2k/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
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% 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]Direct DFT using complex arithmetic in MATLAB
CS3291 : Section 8
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% 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
CS3291 : Section 8
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% 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
CS3291 : Section 8
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// 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
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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'. Dont need to know how its 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
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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
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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
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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
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When N is even, the DTFT, R(ej) of {r[n]} is:Dirichlet Kernel
CS3291 : Section 8
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sincs10(/(2)) plotted against :-
CS3291 : Section 8
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CS3291 : Section 8
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&A
Page &P
w rads/sec
|sincs10(w/2)|
Modulus ofDirichlet Kernel order 10
Chart2
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&A
Page &P
rads/sample
|sincs10(W/(2pi))|
Modulus ofDirichlet Kernel of order 10
Sheet1
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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
-
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
-
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) MultiplicationMultiplication(windowing) Complex convolution
CS3291 : Section 8
-
Duality of time- & frequency-domain convolution Time-domain Frequency-domainConvolution (filtering) MultiplicationMultiplication(windowing) Complex convolution
CS3291 : Section 8
-
(1/2) ( (-0) + (+0) ) ejn 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)Spectral analysis of sampled sine-waves DTFT of {cos(0n)} cannot be found directly. But inverse DTFT of X(e j ) = ( - 0) + ( + 0) is:
CS3291 : Section 8
-
CS3291 : Section 8
-
DTFT of rectangularly windowed sampled sine-wave DTFT of {cos0n}0,N-1 = DTFT of {x[n].r[n]} = X(ej) R(ej) = P(ej) say. By frequency-domain convolution formula,
CS3291 : Section 8
-
0--0X(e j )R(e j( ) )
CS3291 : Section 8
Chart2
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w rads/sample
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sincs20(w/2)
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X(e j ) = ( - 0) + ( + 0)Now draw this & its modulus:
CS3291 : Section 8
-
0- 0P(e j )
CS3291 : Section 8
Chart2
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| P(e j ) | 0 0
CS3291 : Section 8
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&A
Page &P
rad/si
|P ( )|
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3.44-0.2947811880.294781188
3.46-0.09191854190.0919185419
3.480.11656723450.1165672345
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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
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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
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Amplitude 20Fig 1 : Magnitude of 64 pt DFT spectrum of cos(0.7363n)
CS3291 : Section 8
(p)
(p/2)
|X(k)|
15
10
5
k
32
30
20
16
10
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0
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Now consider 64 pt DFT of { cos (n/4) } in Fig 20 = /4 coincides exactly with 8. Only X[8] affected.
CS3291 : Section 8
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Ampl. = 32Approx. 36% differencein ampl. For same ampl. of sinusoid.Fig2 : Magnitude of 64 point DFT of cos(pn/4).
CS3291 : Section 8
(p)
(p/2)
|X(k)|
32
k
32
16
8
0
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What has happened to the sincs function in Figure 2 ? We dont see it because all the samples apart from oneoccur 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
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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
Chart1
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0.0061554042
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w[n]
Hann window
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00.0061558193
10.0544966476
20.1464463749
30.2730043364
40.4217821778
50.5782165118
60.7269944815
70.853552687
80.9455027501
90.9938439731
100.9938443882
110.9455039548
120.8535545633
130.7269968458
140.5782191327
150.4217847987
160.2730067007
170.1464482512
180.0544978523
190.0061562344
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CS3291 : Section 8
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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
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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
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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
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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
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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
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Parsevals 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
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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
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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
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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
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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
