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Ch.4 Fourier Analysis of Discrete-Time Signals Kamen and Heck

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4.1 Discrete-Time Fourier Transform X( ) = n=- , x[n] e -j n ( Eq. 4.1) Complex valued function of real variable , the frequency. A sufficient condition for x[n] to have a DTFT in the ordinary sense is that x[n] be absolutely summable.

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Example 4.1 Computation of the DTFT Consider x[n] = a n, 0 n q and 0 otherwise. The DTFT is X( ) = n=- , x[n] e -j n = n=0,q a n e -j n = n=0,q (ae -j ) n = [1 – (ae -j ) q+1 ] / [1- (ae -j )] (where the closed form expression for a partial sum exponential is used—(Eq.4.5)

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4.1 Discrete-Time Fourier Transform (cont.) X( ) is a periodic function of with period 2 . Rectangular Form: X( ) = R( ) + jI( ). R( ) = n=- , x[n] cos(n ) I( ) = - n=- , x[n] sin(n ) Polar Form: X( ) = |X( )| +exp[j X( )]. –|X( )| = SQRT[R 2 ( ) + I 2 ( )]. – X( )=tan -1 [I( )/ R( )] when R( ) 0 – = + tan -1 [I( )/ R( )] when R( ) < 0

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Example 4.2 Rectangular and Polar Forms –Consider x[n] = a n u(n). –This is similar to Ex. 4.1 except we have q . –Consider the DTFT from Ex. 4.1 but let q : X( ) = lim q [1 – (ae -j ) q+1 ] / [1- (ae -j )] This limit exists for |a| < 1. For this case, the DTFT exists in the ordinary sense. X( ) = 1/ [1- (ae -j )] (Eq. 4.16) The rectangular and polar forms are shown on pages

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4.1.1 Signals with Even or Odd Symmetry Let x[n] be a real-valued discrete-time signal that is an even function (ie, x[n] = x[- n].) –The DTFT is X( )= x[0] + n=1, 2x[n] cos(n ) Let x[n] be an odd function (ie,x[n]=-x[-n]) –The DTFT is X( )= x[0] - n=1, j2x[n]sin(n )

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Example 4.3 DTFT of Rectangular Pulse –Let p[n] = 1 for -q n q and 0 elsewhere. –The signal is even but it is easier to use 4.2. –P( ) = n=-q,q e -j n – =[ e j q – e -j (q+1) ] / [1- e -j ] – = sin[(q + 1/2) ]/[sin( /2)] –This is the discrete-time counterpart to the transform of the rectangular pulse (Ex. 3.9). –Figure 4.3 illustrates the DTFT.

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4.1.2 Spectrum of a Discrete-Time Signal For simplification, the discrete-time Fourier series is not discussed. For a discrete time signal that is not a function of sinusoids the spectrum is a continuum of frequency components. The frequency spectrum is made up of the amplitude spectrum and the phase spectrum. The highest value of = .

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Example 4.4 Decaying Exponential Assume that x[n] = (.5) n u(n). The signal is plotted in Fig The spectrum is shown in Figure 4.2 Note that most of the spectrum is in the lower frequencies.

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Example 4.5 Signal with High- Frequency Components Consider x[n] = (-.5) n u(n). From Figure 4.4 we see that there should be higher frequency components in this signal. From the result of Ex 4.2, the DTFT is: –X( ) = 1/ [1- (-.5e -j )] = 1/ [1 +.5e -j ] –The amplitude and phase spectra are given by equations 4.25 and 4.26 and plotted in Figure 4.5.

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4.1.3 Inverse DTFT x[n] = 1/2 0 2 X( ) e jn d (Eq. 4.7)

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4.1.4 Generalized DTFT Example 4.6 DTFT of a Constant Signal Let x[n] =1 for all n. This signal does not have a DTFT in the ordinary sense—(Why?) Figure 4.6 shows the generalized DTFT. Discussion on page 176 illustrates that its inverse is the constant signal.

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DTFT Transform Pairs and Properties Transform Pairs—Table 4.1 page 177. Properties—Table 4.2 page 178. –No duality property, but there is a relationship between the inverse of the CTFT and the DTFT. –Result can be used to generate DTFT pairs from CTFT pairs—see Example 4.7.

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4.2 Discrete Fourier Transform Let x[n] be a discrete-time signal. Let X( ) is the DTFT of x[n]. Note: the DTFT is a continuous function of . Let N be a positive integer, then the DFT of x[n] is: –X k = n=0,N-1 x[n] e -j2 kn/N, k=0,1,2,…N-1

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4.2 The DFT (p.2) In general, X k is a function of the discrete integer k. There are N values in the DFT of x[n]. These values are complex numbers. Polar form: X k = |X k | exp [j X k ] Rectangular form: X k = R k + jI k –See equations 4.36, MATLAB—program on page 180.

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4.2 The DFT (p.3) Example 4.8 Computation of the DFT –Finite sequence –page Symmetry –Magnitude of the DFT is symmetric about N/2, for N even. –Phase angle of the DFT has odd symmetry about N/2 when N is even Inverse DFT—see equation 4.40 and MATLAB program and Example 4.9 on page 183.

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The DFT (p.4) Sinusoidal Form –The right hand side of the IDFT equation can be written as sinusoids. –See equation 4.45 and Example Relationship to DTFT –If x[n] = 0 for n<0 and n N, the DFT Xk can be viewed as a freqeuency sample version of the DTFT. –X k =X( ) =2 k/N = X(2 k/N ), k = 0,1,2,…,N-1

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Example 4.11 DTFT and DFT of a Pulse Consider p[n] from example 4.3. Let x[n] be p[n-q]. Figure 4.10 shows the amplitude spectrum for q=5. Figure 4.11 shows the amplitude of the DFT for q=5 and N= 22. Figure 4.12 shows the amplitude of the DFT for q=5 and N = 88.

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4.4 FFT Algorithm Consider the DFT and Inverse DFT: X k = Σ n=0,1,…,N-1 x[n] e -j2 kn/N k=0,1,…,N-1 x[n]= (1/N ) Σ k=0,…,N-1 X k e j2 kn/N, n=0,…,N-1 How many multiplications are needed to compute the DFT? (N 2 ) The FFT algorithm requires N(log 2 N)/2 multiplications.

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4.4 FFT Algorithm (p.2) If N = 1024, – DFT requires 1,048,576 multiplications – FFT requires 5,120 multiplications There are different variations of the FFT algorithm. One uses “decimation-in-time”.

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4.4 FFT (p.3) Decimation-in-Time –Subdivide the time interval into intervals having a smaller number of points.

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4.4 FFT (p.4) X k can be broken up into two parts. –First let exp(-j2 /N) = W N –Then X k = Σ n=0,1,…,N-1 x[n]( W N ) kn k=0,1,…,N-1 –Let N be an even integer: a[n]=x[2n] ; b[n]=x[2n + 1], for n = 0,…,N/2. –Let A k = Σ n=0,…,N/2-1 a[n] (W N/2 ) kn, k=0,1,…N/2-1 –Let B k = Σ n=0,…,N/2-1 b[n] (W N/2 ) kn, k=0,1,…N/2-1 –Then X k = A k + (W N ) k B k, k=0,1,…,N/2 -1 –And X (N/2)+k = A k - (W N ) k B k, k=0,1,…,N/2 -1 –See page 197 for the verification.

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4.4 FFT (p.5) Note that the two parts are (N/2) DFTs. This can continue until signals with only one nonzero value are obtained if N is a power of 2. The process is graphically illustrated by Figure To have the outputs in the correct order, a process called bit reversing (see Table 4.3) is used.

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4.4.1 Applications of the FFT Algorithm Computation of the Fourier Transform Convolution Data Analysis –Extraction of a Sinusoidal Component Embedded in Noise –Analysis of Sunspot Data –Stock Price Analysis

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