Question

Consider two 16-point sequences x[n] and h[n]. Let the linear convolution of x[n] and h[n] be denoted by y[n], while z[n] denotes the 16-point Inverse Discrete Fourier Transform (IDFT) of the product of the 16-point DFTs of x[n] and h[n]. The value(s) of k for which z[k] is:

a. k=0,15
b. k=1,14
c. k=2,13
d. k=3,12

Answer 1

To find the value of k for which z[k] is equal to a given set of values, we need to understand the relationship between linear convolution and the IDFT of the product of DFTs.

Step-by-step explanation:

The value(s) of k for which z[k] is:
a. k=0,15
b. k=1,14
c. k=2,13
d. k=3,12

To find the value of k for which z[k] is equal to a given set of values, we need to understand the relationship between linear convolution and the IDFT of the product of DFTs.
Linear convolution: The linear convolution of two sequences x[n] and h[n] is calculated by taking the inverse DFT of the product of the DFTs of x[n] and h[n].
IDFT of product of DFTs: The IDFT of the product of the DFTs of x[n] and h[n] is calculated by taking the product of the DFTs of x[n] and h[n], and then taking the IDFT of the resulting sequence.

Based on this, we can see that the given options represent the possible values of k that would result in the desired values of z[k].
a. k=0,15: This corresponds to the first and last elements of the IDFT sequence.
b. k=1,14: This corresponds to the second and second-to-last elements of the IDFT sequence.
c. k=2,13: This corresponds to the third and third-to-last elements of the IDFT sequence.
d. k=3,12: This corresponds to the fourth and fourth-to-last elements of the IDFT sequence.