Digital Signal Processing Questions and Answers – Linear Filtering Approach to Computation of DFT

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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 log2N, a direct computation of the desired values is more efficient than FFT algorithm.
a) True
b) False
View Answer

Answer: a
Explanation: 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.
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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

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

Answer: d
Explanation: 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
View Answer

Answer: b
Explanation: 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) \(\frac{1}{1+W_N^k z^{-1}}\)
View Answer

Answer: a
Explanation: We know that hk(n)= WN-kn u(n)
On applying z-transform on both sides, we get
Hk(z)=\(\frac{1}{1-W_N^{-k} z^{-1}}\)
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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
View Answer

Answer: b
Explanation: 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
View Answer

Answer: c
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 {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
View Answer

Answer: d
Explanation: 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
View Answer

Answer: c
Explanation: 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).
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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
View Answer

Answer: a
Explanation: 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)
View Answer

Answer: b
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 Mlog2M.

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Manish Bhojasia, a technology veteran with 20+ years @ Cisco & Wipro, is Founder and CTO at Sanfoundry. He is Linux Kernel Developer & SAN Architect and is passionate about competency developments in these areas. He lives in Bangalore and delivers focused training sessions to IT professionals in Linux Kernel, Linux Debugging, Linux Device Drivers, Linux Networking, Linux Storage, Advanced C Programming, SAN Storage Technologies, SCSI Internals & Storage Protocols such as iSCSI & Fiber Channel. Stay connected with him @ LinkedIn