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Note for Digital Signal Processing - DSP By Amity Kumar

  • Digital Signal Processing - DSP
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  • Amity University - AMITY
  • Electronics and Communication Engineering
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DISCRETE FOURIER TRANSFORM 1. Introduction The sampled discrete-time fourier transform (DTFT) of a finite length, discrete-time signal is known as the discrete Fourier transform (DFT). The DFT contains a finite number of samples equal to the number of samples N in the given signal. Computationally efficient algorithms for implementing the DFT go by the generic name of fast Fourier transforms (FFTs). This chapter describes the DFT and its properties, and its relationship to DTFT. 2. Definition of DFT and its Inverse Lest us consider a discrete time signal x (n) having a finite duration, say in the range 0 ≤ n ≤N-1. The DTFT of the signal is N-1 X ( ω) = Σ -jwn x (n)e (1) n-0 Let us sample X using a total of N equally spaced samples in the range : ω ∈(0,2π), sampling interval is 2π That is, we sample X(ω) using the frequencies. N ω= ωk = 2πk , 0 ≤ k ≤N-1. N-1 -jwn Thus X (k) = x (n)e n-0 Σ N-1 X (k) = Σ n-0 x (n)e - j2πkn so the (2) (2) N The result is, by definition the DFT. That is , Equation (0.2) is known as N-point DFT analysis equation. Fig 0.1 shows the Fourier transform of a discrete – time signal and its DFT samples.

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x(w) π 0 →w 2π Fig.1 Sampling of X(w) to get x(k) While working with DFT, it is customary to introduce a complex quantity WN = e-j2π /N Also, it is very common to represent the DFT operation N=1 X(k) = DFT ( x(n)) = Σ x(n) WNkn, 0≤n≤N-1 n=0 The complex quantity Wn is periodic with a period equal to N. That is, WNa+N = e-j+2π/N(a+N) = e-j2π /N n = WNa where a is any integer. Figs. 0.2(a) and (b) shows the sequence odd respectively. 6 5 7 4 for 0≤n≤N-1 in the z-plane for N being even and 5 4 6 0 3 1 2 (a) 0 3 1 2 (b) Fig.2 The Sequence for even N (b) The sequence for odd N. The sequence WkNN for 0 ≤ n ≤ N-1 lies on a circle of unit radius in the complex plane and the phases are equally spaced, beginning at zero. The formula given in the lemma to follow is a useful tool in deriving and analyzing various DFT oriented results.

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2.1. Lemma N-1 Σ Wkn = N δ (k) = { N, k = 0 n-0 N (3) 0, k ≠ 0 Proof : N-1 Σ an = 1 - aN : a≠ 1 n-0 1-a We know that Applying the above result to the left side of equation (3.3), we get N-1 Σ (WkN) n = 1- WkNN n= 0 1- WkNN = 1- e-j2π kNN 1- e-j2π kNN = 1- 1 1- e- j2π kNN = 0, k ≠ 0 : k ≠0 when k = 0, the left side of equation (3.3) becomes N-1 N-1 Σ WN0xn = n= 0 Σ 1=N n =0 Hence, we may write N-1 N, k = 0 Σ WN0xn = 0, k ≠ 0 n=0 = N δ (k), 0≤ k≤ N-1 2.2 Inverse DFT The DFT values (X(k), 0≤ k≤ N-1), uniquely define the sequence x(n) through the inverse DFT formula (IDFT) : N-1 x (n) = IDFT (X(k) = 1 Σ X(k) WN-kn , 0≤ k≤ N-1 N k=0 The above equation is known as the synthesis equation.

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N-1 N-1 N-1 1 Σ X(k) WN-kn = 1 Σ Σ x(m) WNkm N k=0 N k=0 m=0 N-1 N-1 N-1 [ Proof : = ] = WN-kn 1 Σ x(m) Σ WN-(n-m)k N k=0 m=0 [ ] It can be shown that N-1 Σ WN(n-m)k = N, n=m 0, n ≠ m Hence, N-1 1 N Σ x(m) Nδ (n-m) = 1 x Nx (m) N m=n ( sifting property) = x(n) 2.3 Periodicity of X (k) and x (n) The N-point DFT and N-point IDFT are implicit period N. Even though x (n) and X (k) are sequences of length – N each, they can be shown to be periodic with a period N because the exponentials WN±kn in the defining equations of DFT and IDFT are periodic with a period N. For this reason, x (n) and X (k) are called implicit periodic sequences. We reiterate the fact that for finite length sequences in DFT and IDFT analysis periodicity means implicit periodicity. This can be proved as follows : N-1 X (k) = Σ x(n) WNkn N-p=0 ⇒ X (k+N) = Σ x(n) WN(k+N)n Since, WNNn = e-j2πnNn = 1, we get N-1 X (k+N) = Σ x(n) WN-kn n=0 = X (k)

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