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For each case below derive an expression for the DFT <math>Y_M[k],k=0,...,M-1</math> of the signal y[n],n=0,...,M-1 in terms of <math>X_N[k],k=0,...,N-1</math> | For each case below derive an expression for the DFT <math>Y_M[k],k=0,...,M-1</math> of the signal y[n],n=0,...,M-1 in terms of <math>X_N[k],k=0,...,N-1</math> | ||
| − | a. <math>y[n]=e^{j\frac{2\pi}{N}n}x[n],\;\;n=0,...,N-1 | + | a. <math>y[n]=e^{j\frac{2\pi}{N}n}x[n],\;\;n=0,...,N-1</math> |
| − | b. <math>y[n]=\left\{\begin{array}{ll}x[N-1], & n=0,\\ x[n-1], & n=1,...,N-1\end{array} \right. | + | b. <math>y[n]=\left\{\begin{array}{ll}x[N-1], & n=0,\\ x[n-1], & n=1,...,N-1\end{array} \right.</math> |
| − | c. <math>y[n]=\left\{\begin{array}{ll}x[n/2], & \text{ | + | c. <math>y[n]=\left\{\begin{array}{ll}x[n/2], & n \text{ is even},\\ 0, & n \text{ is odd},\end{array} \right. n=0,...,2N-1</math> |
| − | d. <math>y[n]=x[2n],\;\;n=0,...,N/2-1 | + | d. <math>y[n]=x[2n],\;\;n=0,...,N/2-1</math> Assume that N is even. |
Revision as of 20:17, 3 October 2010
Week7 Quiz Pool
Q1: As part of the first stage in a radix 2 FFT, a sequence x[n] of length N = 8 is decomposed into two sequences of length 4 as
$ f_0[n] = x[2n]\text{ , n = 0, 1, 2, 3} $
$ f_1[n] = x[2n + 1]\text{ , n = 0, 1, 2, 3} $
The 4-pt. DFT of each of these two sequences is $ F_0[k]\text{ and }F_1[k] $ respectively.
The specific values of $ F_0[k]\text{ and }F_1[k] $, k = 0, 1, 2, 3, obtained from the length N = 8 sequence in question are listed in the Table below.
From the values of $ F_0[k] \text{ and }F_1[k] $, k = 0, 1, 2, 3, and the values of
$ W_8^k = e^{\frac{-j2\pi k}{8}}, k = 0, 1, 2, 3 $
provided in the Table, determine the numerical values of the actual N = 8-pt. DFT of x[n] denoted $ X_8[k] $ for k = 0, 1, 2, 3, 4, 5, 6, 7.
Answer Here.
Q2: The underlying length N = 8 sequence of x[n] in Q1 may be expressed as
$ x[n] = e^{j2\pi k_1n/8} + e^{j2\pi k_2n/8}, n = 0,1,2 ..., 7 $
Given the values of $ X_8[k] $ for k = 0, 1, ..., 7 determined in Q1, determine the numerical values of $ k_1\text{ and }k_2 $.
Answer Here.
Q3: Let $ X_N[k],k=0,...,N-1 $ denote the N point Discrete Fourier Transform (DFT) of the signal x[n],n=0,...,N-1.
For each case below derive an expression for the DFT $ Y_M[k],k=0,...,M-1 $ of the signal y[n],n=0,...,M-1 in terms of $ X_N[k],k=0,...,N-1 $
a. $ y[n]=e^{j\frac{2\pi}{N}n}x[n],\;\;n=0,...,N-1 $
b. $ y[n]=\left\{\begin{array}{ll}x[N-1], & n=0,\\ x[n-1], & n=1,...,N-1\end{array} \right. $
c. $ y[n]=\left\{\begin{array}{ll}x[n/2], & n \text{ is even},\\ 0, & n \text{ is odd},\end{array} \right. n=0,...,2N-1 $
d. $ y[n]=x[2n],\;\;n=0,...,N/2-1 $ Assume that N is even.
Q4: Consider complex signal $ x[n]=\left\{\begin{array}{ll}e^{j\omega _0 n} & 0<=n<=N-1,\\ 0, & \text{otherwise},\end{array} \right. $
a. Compute $ X(e^{j\omega}) $ the Fourier Transform of x[n].
b. Compute X[k], the N points DFT of x[n].
c. For $ \omega _0=2\pi k_0 /N, $ where $ k_0 $ is integer, compute DFT of x[n].
