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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 -jn (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] = an, 0nq and 0 otherwise. The DTFT is X() = n=-, x[n] e -jn = n=0,q an e -jn = 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[R2() + I2()]. 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] = an 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 -jn =[ e jq – 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. 4.1. 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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Inverse DTFT x[n] = 1/2 02 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**

4.1.5 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: Xk = n=0,N-1 x[n] e -j2kn/N , k=0,1,2,…N-1

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4.2 The DFT (p.2) In general, Xk 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: Xk = |Xk| exp [jXk] Rectangular form: Xk = Rk + jIk See equations 4.36, 4.37. MATLAB—program on page 180.

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**4.2 The DFT (p.3) Example 4.8 Computation of the DFT 4.2.1 Symmetry**

Finite sequence –page 181. 4.2.1 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. 4.2.2 Inverse DFT—see equation 4.40 and MATLAB program and Example 4.9 on page 183.

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**The DFT (p.4) 4.2.3 Sinusoidal Form 4.2.4 Relationship to DTFT**

The right hand side of the IDFT equation can be written as sinusoids. See equation 4.45 and Example 4.10. 4.2.4 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. Xk =X() =2k/N = X(2k/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:**

Xk = Σn=0,1,…,N-1 x[n] e -j2kn/N k=0,1,…,N-1 x[n]= (1/N ) Σk=0,…,N-1 Xk e j2kn/N, n=0,…,N-1 How many multiplications are needed to compute the DFT? (N2) The FFT algorithm requires N(log2N)/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) Xk can be broken up into two parts.**

First let exp(-j2/N) = WN Then Xk = Σn=0,1,…,N-1 x[n]( WN )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 Ak = Σ n=0,…,N/2-1 a[n] (WN/2)kn, k=0,1,…N/2-1 Let Bk = Σ n=0,…,N/2-1 b[n] (WN/2)kn, k=0,1,…N/2-1 Then Xk = Ak + (WN)k Bk, k=0,1,…,N/2 -1 And X(N/2)+k = Ak - (WN)k Bk, 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 4.21. 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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