Fast Fourier Transform Algorithm MCQ Quiz – Objective Question with Answer for Fast Fourier Transform Algorithm

31. How many complex multiplications is required per output data point?

A. [(N/2)logN]/L

B. [Nlog22N]/L

C. [(N/2)log2N]/L

D. None of the mentioned

Answer: B

In the overlap-add method, the N-point data block consists of L new data points and additional M-1 zeros and the number of complex multiplications required in the FFT algorithm are (N/2)log2N. So, the number of complex multiplications per output data point is [Nlog22N]/L.

 

31. If the desired number of values of the DFT is less than log2N, a direct computation of the desired values is more efficient than the FFT algorithm.

A. True
B. False

Answer: A

To calculate an N point DFT using the FFT algorithm, we need to perform (N/2) log2N multiplications and N log2N additions.

But in some cases where the 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 the FFT algorithm.

 

32. 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

The 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 linear filtering of a set of input data. As a consequence, the FFT algorithm can be used to compute the Chirp-z transform.

 

33. 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

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

 

34. 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

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).

 

35. 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

We know that hk(n)= WN-kn u(n)

On applying z-transform on both sides, we get

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

 

36. 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

We know that hk(n)=WN-kn u(n)=yk(n)/x(n)

Hence the expression to compute yk(n) recursively is

=> yk(n)=WN-kyk(n-1)+x(n).

 

37. 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

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

 

38. 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

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

 

39. 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

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).

 

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

A. R0>1
B. R0<1
C. R0=1
D. None of the mentioned

Answer: A

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.

Scroll to Top