This set of Digital Signal Processing Multiple Choice Questions & Answers (MCQs) focuses on “Linear Filtering Approach to Computation of DFT”.

1. If the desired number of values of the DFT is less than log_{2}N, a direct computation of the desired values is more efficient than FFT algorithm.

a) True

b) False

View Answer

Explanation: To calculate a N point DFT using FFT algorithm, we need to perform (N/2) log

_{2}N multiplications and N log

_{2}N additions. But in some cases where desired number of values of the DFT is less than log

_{2}N such a huge complexity is not required. So, direct computation of the desired values is more efficient than FFT algorithm.

2. What is the transform that is suitable for evaluating the z-transform of a set of data on a variety of contours in the z-plane?

a) Goertzel Algorithm

b) Fast Fourier transform

c) Chirp-z transform

d) None of the mentioned

View Answer

Explanation: Chirp-z transform algorithm is suitable for evaluating the z-transform of a set of data on a variety of contours in the z-plane. This algorithm is also formulated as a linear filtering of a set of input data. As a consequence, the FFT algorithm can be used to compute the Chirp-z transform.

3. According to Goertzel Algorithm, if the computation of DFT is expressed as a linear filtering operation, then which of the following is true?

a) y_{k}(n)=\(\sum_{m=0}^N x(m)W_N^{-k(n-m)}\)

b) y_{k}(n)=\(\sum_{m=0}^{N+1} x(m)W_N^{-k(n-m)}\)

c) y_{k}(n)=\(\sum_{m=0}^{N-1} x(m)W_N^{-k(n+m)}\)

d) y_{k}(n)=\(\sum_{m=0}^{N-1} x(m)W_N^{-k(n-m)}\)

View Answer

Explanation: Since W

_{N}

^{-kN}= 1, multiply the DFT by this factor. Thus

X(k)=W

_{N}

^{-kN}\(\sum_{m=0}^{N-1} x(m)W_N^{-km}=\sum_{m=0}^{N-1} x(m)W_N^{-k(N-m)}\)

The above equation is in the form of a convolution. Indeed, we can define a sequence y

_{k}(n) as

y

_{k}(n)=\(\sum_{m=0}^{N-1} x(m)W_N^{-k(n-m)}\)

4. If y_{k}(n) is the convolution of the finite duration input sequence x(n) of length N, then what is the impulse response of the filter?

a) WN-kn

b) WN-kn u(n)

c) WNkn u(n)

d) None of the mentioned

View Answer

Explanation: We know that y

_{k}(n)=\(\sum_{m=0}^{N-1} x(m)W_N^{-k(n-m)}\)

The above equation is of the form y

_{k}(n)=x(n)*h

_{k}(n)

Thus we obtain, h

_{k}(n)= WN

^{-kn}u(n).

5. What is the system function of the filter with impulse response h_{k}(n)?

a) \(\frac{1}{1-W_N^{-k} z^{-1}}\)

b) \(\frac{1}{1+W_N^{-k} z^{-1}}\)

c) \(\frac{1}{1-W_N^k z^{-1}}\)

d) \(\frac{1}{1+W_N^k z^{-1}}\)

View Answer

Explanation: We know that h

_{k}(n)= W

_{N}

^{-kn}u(n)

On applying z-transform on both sides, we get

H

_{k}(z)=\(\frac{1}{1-W_N^{-k} z^{-1}}\)

6. What is the expression to compute y_{k}(n) recursively?

a) y_{k}(n)=W_{N}^{-ky}_{k}(n+1)+x(n)

b) y_{k}(n)=W_{N}^{-ky}_{k}(n-1)+x(n)

c) y_{k}(n)=W_{N}ky_{k}(n+1)+x(n)

d) None of the mentioned

View Answer

Explanation: We know that h

_{k}(n)=W

_{N}

^{-kn}u(n)=y

_{k}(n)/x(n)

=> y

_{k}(n)=W

_{N}

^{-ky}

_{k}(n-1)+x(n).

7. What is the equation to compute the values of the z-transform of x(n) at a set of points {zk}?

a) \(\sum_{n=0}^{N-1} x(n) z_k ^n\), k=0,1,2…L-1

b) \(\sum_{n=0}^{N-1} x(n) z_{-k}^{-n}\), k=0,1,2…L-1

c) \(\sum_{n=0}^{N-1} x(n) z_k^{-n}\), k=0,1,2…L-1

d) None of the mentioned

View Answer

Explanation: According to the Chirp-z transform algorithm, if we wish to compute the values of the z-transform of x(n) at a set of points {z

_{k}}. Then,

X(z

_{k})=\(\sum_{n=0}^{N-1} x(n) z_k^{-n}\), k=0,1,2…L-1

8. If the contour is a circle of radius r and the z_{k} are N equally spaced points, then what is the value of z_{k}?

a) re^{-j2πkn/N}

b) re^{jπkn/N}

c) re^{j2πkn}

d) re^{j2πkn/N}

View Answer

Explanation: We know that, if the contour is a circle of radius r and the z

_{k}are N equally spaced points, then what is the value of z

_{k}is given by re

^{j2πkn/N}

9. How many multiplications are required to calculate X(k) by chirp-z transform if x(n) is of length N?

a) N-1

b) N

c) N+1

d) None of the mentioned

View Answer

Explanation: We know that y

_{k}(n)=W

_{N}

^{-k}y

_{k}(n-1)+x(n).Each iteration requires one multiplication and two additions. Consequently, for a real input sequence x(n), this algorithm requires N+1 real multiplications to yield not only X(k) but also, due to symmetry, the value of X(N-k).

10. If the contour on which the z-transform is evaluated is as shown below, then which of the given condition is true?

a) R_{0}>1

b) R_{0}<1

c) R_{0}=1

d) None of the mentioned

View Answer

Explanation: From the definition of chirp z-transform, we know that V=R

_{0}e

^{jθ}.

If R

_{0}>1, then the contour which is used to calculate z-transform is as shown below.

11. How many complex multiplications are need to be performed to calculate chirp z-transform?(M=N+L-1)

a) log_{2}M

b) Mlog_{2}M

c) (M-1)log_{2}M

d) Mlog_{2}(M-1)

View Answer

Explanation: Since we will compute the convolution via the FFT, let us consider the circular convolution of the N point sequence g(n) with an M point section of h(n) where M>N. In such a case, we know that the first N-1 points contain aliasing and that the remaining M-N+1 points are identical to the result that would be obtained from a linear convolution of h(n) with g(n). In view of this, we should select a DFT of size M=L+N-1. Thus the total number of complex multiplications to be performed are Mlog

_{2}M.

**Sanfoundry Global Education & Learning Series – Digital Signal Processing.**

To practice all areas of Digital Signal Processing, __here is complete set of 1000+ Multiple Choice Questions and Answers__.