1. By means of the DFT and IDFT, determine the response of the FIR filter with impulse response h(n)={1,2,3} to the input sequence x(n)={1,2,2,1}?

A. {1,4,11,9,8,3}
B. {1,4,9,11,8,3}
C. {1,4,9,11,3,8}
D. {1,4,9,3,8,11}

Answer: B

The input sequence has a length N=4 and the impulse response has a length M=3. So, the response must have a length of 6(4+3-1).

We know that, Y(k)=X(k).H(k)

Thus we obtain Y(k)={36,-14.07-j17.48,j4,0.07+j0.515,0,0.07-j0.515,-j4,-14.07+j17.48}

By applying IDFT to the above sequence, we get y(n)={1,4,9,11,8,3,0,0}

Thus the output of the system is {1,4,9,11,8,3}.

2. What is the sequence y(n) that results from the use of four point DFTs if the impulse response is h(n)={1,2,3} and the input sequence x(n)={1,2,2,1}?

A. {9,9,7,11}
B. {1,4,9,11,8,3}
C. {7,9,7,11}
D. {9,7,9,11}

Answer: D

The four point DFT of h(n) is H(k)=1+2e-jkπ/2+3 e-jkπ (k=0,1,2,3)

Hence H(0)=6, H(1)=-2-j2, H(3)=2, H(4)=-2+j2

The four point DFT of x(n) is X(k)= 1+2e-jkπ/2+2 e-jkπ+3e-3jkπ/2(k=0,1,2,3)

Hence X(0)=6, X(1)=-1-j, X(2)=0, X(3)=-1+j

The product of these two four point DFTs is

Ŷ(0)=36, Ŷ(1)=j4, Ŷ(2)=0, Ŷ(3)=-j4

The four point IDFT yields ŷ(n)={9,7,9,11}

We can verify as follows

We know that from the previous question y(n)={1,4,9,11,8,3}

3. Overlap add and Overlap save are the two methods for linear FIR filtering a long sequence on a block-by-block basis using DFT.

A. True
B. False

Answer: A

In these two methods, the input sequence is segmented into blocks and each block is processed via DFT and IDFT to produce a block of output data.

The output blocks are fitted together to form an overall output sequence that is identical to the sequence obtained if the long block had been processed via time-domain convolution.

So, Overlap adds and Overlap save are the two methods for linear FIR filtering a long sequence on a block-by-block basis using DFT.

4. In the Overlap save method of long sequence filtering, what is the length of the input sequence block?

A. L+M+1
B. L+M
C. L+M-1
D. None of the mentioned

Answer: C

In this method, each data block consists of the last M-1 data points of the previous data block followed by L new data points to form a data sequence of length N=L+M-1.

5. In the Overlap save method of long sequence filtering, how many zeros are appended to the impulse response of the FIR filter?

A. L+M
B. L
C. L+1
D. L-1

Answer: D

The impulse of the FIR filter is increased in length by appending L-1 zeros and an N-point DFT of the sequence is computed once and stored.

6. The first M-1 values of the output sequence in every step of the Overlap save method of filtering of long sequence are discarded.

A. True
B. False

Answer: A

Since the data record of length N, the first M-1 points of ym(n) are corrupted by aliasing and must be discarded. The last L points of ym(n) are exactly as same as the result from linear convolution.

7. In the Overlap add method, what is the length of the input data block?

A. L-1
B. L
C. L+1
D. None of the mentioned

Answer: B
In this method, the size of the input data block is L points and the size of the DFTs and IDFT is N=L+M-1.

8. Which of the following is true in the case of the Overlap add method?

A. M zeros are appended at last of each data block
B. M zeros are appended at first of each data block
C. M-1 zeros are appended at last of each data block
D. M-1 zeros are appended at first of each data block

Answer: C

In the Overlap add method, to each data block we append M-1 zeros at last and compute N point DFT so that the length of the input sequence is L+M-1=N.

9. In which of the following methods, the input sequence is considered as shown in the below diagram?

A. Overlap save method
B. Overlap add method
C. Overlap add & save method
D. None of the mentioned

Answer: A
From the figure given, we can notice that each data block consists of the last M-1 data points of the previous data block followed by L new data points to form a data sequence of length N+L+M-1 which is the same as in the case of Overlap save method.

10. In which of the following methods, the output sequence is considered as shown in the below diagram?

A. Overlap save method
B. Overlap add method
C. Overlap add & save method
D. None of the mentioned

Answer: B

From the figure given, it is clear that the last M-1 points of the first sequence and the first M-1 points of the next sequence are added and nothing is discarded because there is no aliasing in the input sequence. This is the same as in the case of the Overlap add method.