# Finite Impulse Response (FIR) System Structure MCQ Quiz – Objective Question with Answer for FIR System Structure

1. In general, an FIR system is described by the difference equation
y(n)=$$\sum_{k=0}^{M-1}b_k x(n-k)$$.

A. True
B. False

The difference equation y(n)=$$\sum_{k=0}^{M-1}b_k x(n-k)$$ describes the FIR system.

2. What is the general system function of an FIR system?

A. $$\sum_{k=0}^{M-1}b_k x(n-k)$$

B. $$\sum_{k=0}^M b_k z^{-k}$$

C. $$\sum_{k=0}^{M-1}b_k z^{-k}$$

D. None of the mentioned

We know that the difference equation of an FIR system is given by

y(n)=$$\sum_{k=0}^{M-1}b_k x(n-k)$$.

=>h(n)=bk=>$$\sum_{k=0}^{M-1}b_k z^{-k}$$.

3. Which of the following is a method for implementing an FIR system?

A. Direct form
C. Lattice structure
D. All of the mentioned

There are several structures for implementing an FIR system, beginning with the simplest structure, called the direct form. There are several other methods like cascade form realization, frequency sampling realization, and lattice realization which are used for implementing an FIR system.

4. How many memory locations are used for storage of the output point of a sequence of length M in direct form realization?

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

The direct form realization follows immediately from the non-recursive difference equation given by

y(n)=$$\sum_{k=0}^{M-1}b_k x(n-k)$$.

We observe that this structure requires M-1 memory locations for storing the M-1 previous inputs.

5. The direct form realization is often called a transversal or tapped-delay-line filter.

A. True
B. False

The structure of the direct form realization resembles a tapped delay line or a transversal system.

6. What is the condition of M, if the structure according to the direct form is as follows?

A. M even
B. M odd
C. All values of M
D. Doesn’t depend on the value of M

When the FIR system has a linear phase, the unit sample response of the system satisfies either the symmetry or asymmetry condition, h(n)=±h(M-1-n)

For such a system the number of multiplications is reduced from M to M/2 for M even and to (M-1)/2 for M odd. Thus for the structure given in the question, M is odd.

7. By combining two pairs of poles to form a fourth-order filter section, by what factor we have reduced the number of multiplications?

A. 25%
B. 30%
C. 40%
D. 50%

We have to do 3 multiplications for every second-order equation. So, we have to do 6 multiplications if we combine two second-order equations and we have to perform 3 multiplications by directly calculating the fourth-order equation. Thus the number of multiplications is reduced by a factor of 50%.

8. The desired frequency response is specified at a set of equally spaced frequencies defined by the equation?

A. $$\frac{\pi}{2M}$$(k+α)

B. $$\frac{\pi}{M}$$(k+α)

C. $$\frac{2\pi}{M}$$(k+α)

D. None of the mentioned

To derive the frequency sampling structure, we specify the desired frequency response at a set of equally spaced frequencies, namely

ωk=$$\frac{2\pi}{M}$$(k+α), k=0,1…(M-1)/2 for M odd

k=0,1….(M/2)-1 for M even

α=0 or 0.5.

9. The realization of the FIR filter by frequency sampling realization can be viewed as a cascade of how many filters?

A. Two
B. Three
C. Four
D. None of the mentioned

In frequency sampling realization, the system function H(z) is characterized by the set of frequency samples {H(k+ α)} instead of {h(n)}. We view this FIR filter realization as a cascade of two filters. One is an all-zero or a comb filter and the other consists of a parallel bank of single-pole filters with resonant frequencies.

10. What is the system function of all-zero filter or comb filter?

A. $$\frac{1}{M}(1+z^{-M} e^{j2πα})$$

B. $$\frac{1}{M}(1+z^M e^{j2πα})$$

C. $$\frac{1}{M}(1-z^M e^{j2πα})$$

D. $$\frac{1}{M}(1-z^{-M} e^{j2πα})$$

The system function H(z) which is characterized by the set of frequency samples is obtained as

H(z)=$$\frac{1}{M}(1-z^{-M} e^{j2πα})\sum_{k=0}^{M-1}\frac{H(k+α)}{1-e^{j2π(k+α)/M} z^{-1}}$$

We view this FIR realization as a cascade of two filters, H(z)=H1(z).H2(z)

Here H1(z) represents the all-zero filter or comb filter whose system function is given by the equation

H1(z)=$$\frac{1}{M}(1-z^{-M} e^{j2πα})$$.

11. The zeros of the system function of the comb filter are located at _______

A. Inside the unit circle
B. On unit circle
C. Outside unit circle
D. None of the mentioned

The system function of the comb filter is given by the equation

H1(z)=$$\frac{1}{M}(1-z^{-M} e^{j2πα})$$

Its zeros are located at equally spaced points on the unit circle at

zk=ej2π(k+α)/M k=0,1,2….M-1

12. What is the system function of the second filter other than comb filter in the realization of FIR filter?

A. $$\sum_{k=0}^M \frac{H(k+α)}{1-e^{\frac{j2π(k+α)}{M}} z^{-1}}$$

B. $$\sum_{k=0}^{M-1} \frac{H(k+α)}{1+e^{\frac{j2π(k+α)}{M}} z^{-1}}$$

C. $$\sum_{k=0}^{M-1} \frac{H(k+α)}{1-e^{\frac{j2π(k+α)}{M}} z^{-1}}$$

D. None of the mentioned

The system function H(z) which is characterized by the set of frequency samples is obtained as

H(z)=$$\frac{1}{M}(1-z^{-M} e^{j2πα})\sum_{k=0}^{M-1}\frac{H(k+α)}{1-e^{\frac{j2π(k+α)}{M}}z^{-1}}$$

We view this FIR realization as a cascade of two filters, H(z)=H1(z).H2(z)
Here H1(z) represents the all-zero filter or comb filter, and the system function of the other filter is given by the equation

H2(z)=$$\sum_{k=0}^{M-1} \frac{H(k+α)}{1-e^{\frac{j2π(k+α)}{M}} z^{-1}}$$

13. Where do the poles of the system function of the second filter locate?

A. ej2π(k+α)M
B. ej2π(k+α)/M
C. ej2π(k-α)/M
D. ejπ(k+α)/M

The system function of the second filter in the cascade of an FIR realization by frequency sampling method is given by

H2(z)=$$\sum_{k=0}^{M-1} \frac{H(k+α)}{1-e^{\frac{j2π(k+α)}{M}} z^{-1}}$$

We obtain the poles of the above system function by equating the denominator of the above equation to zero.

=>$$1-e^{\frac{j2π(k+α)}{M}} z^{-1}$$=0

=>z=pk=$$e^{\frac{j2π(k+α)}{M}}$$, k=0,1….M-1

14. When the desired frequency response characteristic of the FIR filter is narrowband, most of the gain parameters {H(k+α)} are zero.

A. True
B. False

When the desired frequency response characteristic of the FIR filter is narrowband, most of the gain parameters {H(k+α)} are zero. Consequently, the corresponding resonant filters can be eliminated and only the filters with nonzero gains need to be retained.

15. Which of the following is the application of lattice filter?

A. Digital speech processing
C. Electroencephalogram
D. All of the mentioned

Lattice filters are used extensively in digital signal processing and in the implementation of adaptive filters.

16. If we consider a sequence of FIR filers with system function Hm(z)=Am(z), then what is the definition of the polynomial Am(z)?

A. $$1+\sum_{k=0}^m α_m (k)z^{-k}$$

B. $$1+\sum_{k=1}^m α_m (k)z^{-k}$$

C. $$1+\sum_{k=1}^m α_m (k)z^k$$

D. $$\sum_{k=0}^m α_m (k)z^{-k}$$

Consider a sequence of FIR filer with system function Hm(z)=Am(z), m=0,1,2…M-1
where, by definition, Am(z) is the polynomial

Am(z)=$$1+\sum_{k=1}^m α_m (k)z^{-k}$$, m≥1 and A0(z)=1.

17. What is the unit sample response of the mth filter?

A. hm(0)=0 and hm(k)=αm(k), k=1,2…m
B. hm(k)=αm(k), k=0,1,2…m(αm(0)≠1)
C. hm(0)=1 and hm(k)=αm(k), k=1,2…m
D. none of the mentioned

We know that Hm(z)=Am(z) and Am(z) is a polynomial whose equation is given as

Am(z)=$$1+\sum_{k=1}^m α_m (k)z^{-k}$$, m≤1 and A0(z)=1

A0(z)=1 => hm(0)=1 and

Am(z)=$$\sum_{k=1}^m α_m (k)z^{-k}$$(m≤1)=> hm(k)=αm(k) for k=1,2…m.

18. The FIR filter whose direct form structure is as shown below is a prediction error filter.

A. True
B. False

The FIR structure shown in the above figure is intimately related to the topic of linear prediction. Thus the top filter structure shown in the above figure is called a prediction error filter.

19. What is the output of the single stage lattice filter if x(n) is the input?

A. x(n)+Kx(n+1)
B. x(n)+Kx(n-1)
C. x(n)+Kx(n-1)+Kx(n+1)
D. Kx(n-1)

The single-stage lattice filter is shown below.

Here both the inputs are excited and the output is selected from the top branch.

Thus the output of the single-stage lattice filter is given by y(n)= x(n)+Kx(n-1).

20. What is the output from the second-stage lattice filter when two single-stage lattice filers are cascaded with an input of x(n)?

A. K1K2x(n-1)+K2x(n-2)
B. x(n)+K1x(n-1)
C. x(n)+K1K2x(n-1)+K2x(n-2)
D. x(n)+K1(1+K2)x(n-1)+K2x(n-2)

When two single stage lattice filters are cascaded, then the output from the first filter is given by the equation

f1(n)= x(n)+K1x(n-1)
g1(n)=K1x(n)+x(n-1)

The output from the second filter is obtained as

f2(n)=f1(n)+K2g1(n-1)
=x(n)+K2[K1x(n-1)+x(n-2)]+ K1x(n-1)
= x(n)+K1(1+K2)x(n-1)+K2x(n-2).

21. What is the value of the coefficient α2(1) in the case of FIR filter represented in direct form structure with m=2 in terms of K1 and K2?

A. K1(K2)
B. K1(1-K2)
C. K1(1+K2)
D. None of the mentioned

The equation for the output of an FIR filter represented in the direct form structure is given as
y(n)=x(n)+ α2(1)x(n-1)+ α2(2)x(n-2)

The output from the double stage lattice structure is given by the equation,
f2(n)= x(n)+K2(1+K2)x(n-1)+K2x(n-2)
By comparing the coefficients of both the equations, we get

α2(1)= K1(1+K2).

22. The constants K1 and K2 of the lattice structure are called reflection coefficients.

A. True
B. False

The equation of the output from the second stage lattice filter is given by
f2(n)= x(n)+K1(1+K2)x(n-1)+K2x(n-2)
In the above equation, the constants K1 and K2 are called reflection coefficients.

23. If a three-stage lattice filter with coefficients K1=1/4, K2=1/2 K3=1/3, then what are the FIR filter coefficients for the direct form structure?

A. (1,8/24,5/8,1/3)
B. (1,5/8,13/24,1/3)
C. (1/4,13/24,5/8,1/3)
D. (1,13/24,5/8,1/3)

We get the output from the third stage lattice filter as
A3(z)=1+(13/24)z-1+(5/8)z-2+(1/3)z-3.

Thus the FIR filter coefficients for the direct form structure are (1,13/24,5/8,1/3).

24. What are the lattice coefficients corresponding to the FIR filter with system function H(z)= 1+(13/24)z-1+(5/8)z-2+(1/3)z-3?

A. (1/2,1/4,1/3)
B. (1,1/2,1/3)
C. (1/4,1/2,1/3)
D. None of the mentioned