*Discrete Convolution and Correlation Viva Questions*

Discrete Convolution and Correlation Viva Questions, Viva Questions on Discrete Convolution and Correlation, Digital Signal Processing Viva Questions, Discrete Convolution Viva Questions, Discrete Correlation Viva Questions, Engineering Viva Questions,

Discrete-Time Signals and Systems Viva Questions

### Short Questions with Answer

**Q.1. ***What is discrete convolution?*

** Ans. **The convolution of discrete-time signals is called discrete convolution.

Discrete-time convolution is a method of finding the zero-state response of relaxed linear time-invariant systems.

**Q.2. ***Write the expression for discrete convolution.*

** Ans. **The expression for discrete convolution

*y*(

*n*) of two discrete-time signals

*x*(

*n*) and

*h*(

*n*) is:

y(n)=x(n)*h(n)=\sum_{k=-\infty }^{\infty }x(k)h(n-k) or

y(n)=h(n)*x(n)=\sum_{k=-\infty }^{\infty }h(k)x(n-k)**Q.3. ***If y(n) = x(n) * h(n), how are the parameters of y(n) related to the parameters of*

*x**(n) and h(n).*

** Ans. **If

*y*(

*n*) =

*x*(

*n*) *

*h*(

*n*), then

(i) The starting index of *y*(*n*) equals the sum of the starting indices of *x*(*n*) and *h*(*n*).

(ii) The ending index of *y*(*n*) equals the sum of the ending indices of *x*(*n*) and *h*(*n*).

(iii) The length *Ly *of *y*(*n*) is related to the lengths *Lx *and *Lh *of *x*(*n*) and *h*(*n*) by *Ly *= *Lx *+ *Lh *– 1.

**Q.4. ***What are the methods to compute the convolution sum of two sequences?*

** Ans. **The methods to compute the convolution sum of two sequences are:

- Graphical method
- Tabular array method
- Tabular method
- Matrices method
- Sum-by-column method
- The flip, shift, multiply and sum method.

**Q.5. ***Write the expressions for convolution sum of causal and non-causal systems excited by causal and non-causal inputs.*

** Ans. **If

*x*(

*n*) is the input and

*h*(

*n*) is the impulse response of the system, the expression for convolution

*y*(

*n*) is obtained as follows:

For a non-causal system excited by a non-causal input,

y(n)=\sum_{k=-\infty }^{\infty }x(k)h(n-k)=\sum_{k=-\infty }^{\infty }h(k)x(n-k)For a non-causal system excited by a causal input,

y(n)=\sum_{k= 0 }^{\infty }x(k)h(n-k)=\sum_{k=-\infty }^{n }h(k)x(n-k)For a causal system excited by a non-causal input,

y(n)=\sum_{k=-\infty }^{n }x(k)h(n-k)=\sum_{k=0 }^{\infty }h(k)x(n-k)For a causal system excited by a causal input,

y(n)=\sum_{k= 0 }^{n }x(k)h(n-k)=\sum_{k= 0 }^{n }h(k)x(n-k)**Q.6. ***Write the properties of discrete convolution.*

** Ans. **The properties of discrete convolution are:

- Commutative property:
*x*(*n*) **h*(*n*) =*h*(*n*) **x*(*n*) - Associative property: [
*x*(*n*) **h*_{1}(*n*)] **h*_{2}(*n*) =*x*(*n*) * [*h*_{1}(*n*) **h*_{2}(*n*)] - Distributive property:
*x*(*n*) * [*h*_{1}(*n*) +*h*_{2}(*n*)] =*x*(*n*) **h*_{1}(*n*) +*x*(*n*) **h*_{2}(*n*) - Shifting property:
*x*(*n -k*) **h*(*n*–*m*) =*y*(*n*–*k*–*m*) - Convolution with an impulse
*x*(*n*) * δ(*n*) =*x*(*n*)

**Q.7. ***What do you mean by flipping a sequence?*

** Ans. **Flipping a sequence means time reversing the sequence.

**Q.8. ***How do you find regular convolution using circular convolution?*

** Ans. **The linear (or regular) convolution of two sequences

*x*(

*n*) (with length

*N*

_{1}) and

*h*(

*n*) (with length

*N*

_{2}) can be found using the periodic convolution by zero-padding the sequences

*x*(

*n*) and

*h*(

*n*) to a length

*N*

_{1}+

*N*

_{2}– 1 and then finding periodic convolution. The regular convolution of original unpadded sequences equals the periodic convolution of the zero-padded sequences.

**Q.9. ***What do you mean by deconvolution?*

** Ans. **If the convolution of

*x*(

*n*) and

*h*(

*n*) is

*y*(

*n*),

*i.e.*

*y*(

*n*) =

*x*(

*n*) *

*h*(

*n*), then deconvolution is a process of finding

*h*(

*n*) (or

*x*(

*n*)) from

*y*(

*n*) for a given

*x*(

*n*) (or

*h*(

*n*)).

**Q.10. ***What are the methods of finding deconvolution?*

** Ans. **Deconvolution may be found using the following methods:

- Polynomial division method
- Recursion method
- Tabular method.

**Q.11. ***What is the basic difference between linear and circular convolution?*

** Ans. **The linear convolution and circular convolution basically involve the same four steps, namely folding one sequence, shifting the folded sequence, multiplying the two sequences and finally summing the value of the product sequences. The difference between the two is that in circular convolution, the folding and shifting (rotating) operations are performed in a circular fashion by computing the index of one of the sequences by modulo-

*N*operation. In linear convolution, there is no modulo-

*N*operation.

**Q.12. ***What are the methods of finding circular convolution?*

** Ans. **The methods of finding circular convolution are:

- Concentric circle method (Graphical method)
- Tabular array method
- Matrices method
- DFT method

**Q.13. ***How do you obtain a periodic extension of a non-periodic signal?*

** Ans. **For finite-length sequences, one way of finding one period of the periodic extension is to wrap-around

*N*-sample sections of

*x*(

*n*) and add them all up. If

*x*(

*n*) is shorter than

*N*, one period of its periodic extension is obtained simply by padding

*x*(

*n*) with enough zeros to increase its length to

*N.*

**Q.14. ***How do you obtain periodic convolution from linear convolution?*

** Ans. **To find periodic convolution from linear convolution, first, make the sequences be of equal length

*N*, find the linear convolution of one period of the sequences which will have (2

*N*– 1) samples, then extend its length to 2

*N*(by appending a zero), slice it into two halves (of length

*N*each), line up the second half with the first, and add the two halves to get the periodic convolution.

**Q.15. ***How do you find the system response to periodic inputs?*

** Ans. **The system response to periodic inputs with period

*N*is obtained by finding the linear convolution of input

*x*(

*n*) and impulse response

*h*(

*n*) and ignoring the first period.

Another approach is to find the output for one period of the input (using regular convolution) and find one period of the periodic output by superposition (using a periodic extension).

One more approach is to first find one period of the periodic extension of the impulse response, then find its regular convolution with one period of the input, and finally, wrap around the regular convolution to generate one period of the periodic output.

**Q.16. ***What is correlation?*

** Ans. **Correlation is a measure of similarity between two signals and is found using a process similar to convolution. The correlation of two signals is equal to the convolution of one signal with the flipped version of the second signal.

**Q.17. ***What is cross-correlation?*

** Ans. **The correlation of two different signals is called cross-correlation.

**Q.18. ***What is autocorrelation?*

** Ans. **The correlation of a signal with itself is called autocorrelation. It gives a measure of similarity between a sequence and its shifted version.

**Q.19. ***What is the use of correlation?*

** Ans. **Correlation is an effective method of detecting signals buried in noise.

**Q.20. ***Where does the autocorrelation function attain its maximum value?*

** Ans. **The autocorrelation function attains its maximum value at the origin, i.e. at

*n*= 0.