250+ TOP MCQs on Linear Filtering Approach to Computation of DFT & Answers

Digital Signal Processing Multiple Choice Questions on “Linear Filtering Approach to Computation of DFT”.

1. If the desired number of values of the DFT is less than log2N, a direct computation of the desired values is more efficient than FFT algorithm.
A. True
B. False
Answer: A
Clarification: To calculate a N point DFT using FFT algorithm, we need to perform (N/2) log2N multiplications and N log2N additions. But in some cases where desired number of values of the DFT is less than log2N 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
Answer: C
Clarification: 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. yk(n)=(sum_{m=0}^N x(m)W_N^{-k(n-m)})
B. yk(n)=(sum_{m=0}^{N+1} x(m)W_N^{-k(n-m)})
C. yk(n)=(sum_{m=0}^{N-1} x(m)W_N^{-k(n+m)})
D. yk(n)=(sum_{m=0}^{N-1} x(m)W_N^{-k(n-m)})
Answer: D
Clarification: Since WN-kN = 1, multiply the DFT by this factor. Thus
X(k)=WN-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 yk(n) as
yk(n)=(sum_{m=0}^{N-1} x(m)W_N^{-k(n-m)})

4. If yk(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
Answer: B
Clarification: We know that yk(n)=(sum_{m=0}^{N-1} x(m)W_N^{-k(n-m)})
The above equation is of the form yk(n)=x(n)*hk(n)
Thus we obtain, hk(n)= WN-kn u(n).

5. What is the system function of the filter with impulse response hk(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}})
Answer: A
Clarification: We know that hk(n)= WN-kn u(n)
On applying z-transform on both sides, we get
Hk(z)=(frac{1}{1-W_N^{-k} z^{-1}})

6. What is the expression to compute yk(n) recursively?
A. yk(n)=WN-kyk(n+1)+x(n)
B. yk(n)=WN-kyk(n-1)+x(n)
C. yk(n)=WNkyk(n+1)+x(n)
D. None of the mentioned
Answer: B
Clarification: We know that hk(n)=WN-kn u(n)=yk(n)/x(n)
=> yk(n)=WN-kyk(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
Answer: C
Clarification: 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 {zk}. Then,
X(zk)=(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 zk are N equally spaced points, then what is the value of zk?
A. re-j2πkn/N
B. rejπkn/N
C. rej2πkn
D. rej2πkn/N
Answer: D
Clarification: We know that, if the contour is a circle of radius r and the zk are N equally spaced points, then what is the value of zk is given by rej2πkn/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
Answer: C
Clarification: We know that yk(n)=WN-kyk(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?
digital-signal-processing-questions-answers-linear-filtering-approach-computation-dft-q10
A. R0>1
B. R0<1
C. R0=1
D. None of the mentioned
Answer: A
Clarification: From the definition of chirp z-transform, we know that V=R0e.
If R0>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. log2M
B. Mlog2M
C. (M-1)log2M
D. Mlog2(M-1)
Answer: B
Clarification: 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 Mlog2M.