# IIR Filter Design by the Bilinear Transformation MCQ Quiz – Objective Question with Answer for IIR Filter Design by the Bilinear Transformation

1. In IIR Filter design by the Bilinear Transformation, the Bilinear Transformation is a mapping from

A. Z-plane to S-plane
B. S-plane to Z-plane
C. S-plane to J-plane
D. J-plane to Z-plane

From the equation,

S=$$\frac{2}{T}(\frac{1-z^{-1}}{1+z^{-1}})$$ it is clear that transformation occurs from s-plane to z-plane

2. In Bilinear Transformation, aliasing of frequency components is been avoided.

A. True
B. False

The bilinear transformation is a conformal mapping that transforms the jΩ-axis into the unit circle in the z-plane only once, thus avoiding the aliasing.

3. Is when compared to other design techniques?

A. True
B. False

IIR Filter Design by Bilinear Transformation is the advanced technique because, in other techniques, only lowpass filters and a limited class of bandpass filters are been supported. But this technique overcomes the limitations of other techniques and supports more.

4. The approximation of the integral in y(t) = $$\int_{t_0}^t y'(τ)dt+y(t_0)$$ by the Trapezoidal formula at t = nT and t0=nT-T yields equation?

A. y(nT) = $$\frac{T}{2} [y^{‘} (nT)+y^{‘} (T-nT)]+y(nT-T)$$

B. y(nT) = $$\frac{T}{2} [y^{‘} (nT)+y^{‘} (nT-T)]+y(nT-T)$$

C. y(nT) = $$\frac{T}{2} [y^{‘} (nT)+y^{‘} (T-nT)]+y(T-nT)$$

D. y(nT) = $$\frac{T}{2} [y^{‘} (nT)+y^{‘} (nT-T)]+y(T-nT)$$

By integrating the equation,

y(t) = $$\int_{t_0}^t y^{‘} (τ)dt+y(t_0)$$ at t=nT and t0=nT-T we get equation,

y(nT) = $$\frac{T}{2} [y^{‘} (nT)+y^{‘} (nT-T)]+y(nT-T)$$.

5. We use y{‘}(nT)=-ay(nT)+bx(nT) to substitute for the derivative in y(nT) = $$\frac{T}{2} [y^{‘} (nT)+y^{‘} (nT-T)]+y(nT-T)$$ and thus obtain a difference equation for the equivalent discrete-time system. With y(n) = y(nT) and x(n) = x(nT), we obtain the result as of the following?

A. $$(1+\frac{aT}{2})Y(z)-(1-\frac{aT}{2})y(n-1)=\frac{bT}{2} [x(n)+x(n-1)]$$

B. $$(1+\frac{aT}{n})Y(z)-(1-\frac{aT}{n})y(n-1)=\frac{bT}{n} [x(n)+x(n-1)]$$

C. $$(1+\frac{aT}{2})Y(z)+(1-\frac{aT}{2})y(n-1)=\frac{bT}{2} (x(n)-x(n-1))$$

D. $$(1+\frac{aT}{2})Y(z)+(1-\frac{aT}{2})y(n-1)=\frac{bT}{2} (x(n)+x(n+1))$$

When we substitute the given equation in the derivative of another we get the resultant required equation.

6. The z-transform of below difference equation is?

$$(1+\frac{aT}{2})Y(z)-(1-\frac{aT}{2})y(n-1)=\frac{bT}{2} [x(n)+ x(n-1)]$$

A. $$(1+\frac{aT}{2})Y(z)-(1-\frac{aT}{2}) z^{-1} Y(z)=\frac{bT}{2} (1+z^{-1})X(z)$$

B. $$(1+\frac{aT}{n})Y(z)-(1-\frac{aT}{2}) z^{-1} Y(z)=\frac{bT}{n} (1+z^{-1})X(z)$$

C. $$(1+\frac{aT}{2})Y(z)+(1-\frac{aT}{n}) z^{-1} Y(z)=\frac{bT}{2} (1+z^{-1})X(z)$$

D. $$(1+\frac{aT}{2})Y(z)-(1+\frac{aT}{2}) z^{-1} Y(z)=\frac{bT}{2} (1+z^{-1})X(z)$$

By performing the z-transform of the given equation, we get the required output/equation.

7. What is the system function of the equivalent digital filter? H(z) = Y(z)/X(z) = ?

A. $$\frac{(\frac{bT}{2})(1+z^{-1})}{1+\frac{aT}{2}-(1-\frac{aT}{2}) z^{-1}}$$

B. $$\frac{(\frac{bT}{2})(1-z^{-1})}{1+\frac{aT}{2}-(1+\frac{aT}{2}) z^{-1}}$$

C. $$\frac{b}{\frac{2}{T}(\frac{1-z^{-1}}{1+z^{-1}}+A.}$$

D. $$\frac{(\frac{bT}{2})(1-z^{-1})}{1+\frac{aT}{2}-(1+\frac{aT}{2}) z^{-1}}$$ & $$\frac{b}{\frac{2}{T}(\frac{1-z^{-1}}{1+z^{-1}}+A.}$$

As we considered analog linear filter with system function H(s) = b/s+a
Hence, we got an equivalent system function

where, s = $$\frac{2}{T}(\frac{1-z^{-1}}{1+z^{-1}})$$.

8. In the Bilinear Transformation mapping, which of the following are correct?

A. All points in the LHP of s are mapped inside the unit circle in the z-plane
B. All points in the RHP of s are mapped outside the unit circle in the z-plane
C. All points in the LHP & RHP of s are mapped inside & outside the unit circle in the z-plane
D. None of the mentioned

The bilinear transformation is a conformal mapping that transforms the jΩ-axis into the unit circle in the z-plane and all the points are linked as mentioned above.

9. In Nth order differential equation, the characteristics of bilinear transformation, let z=rejw,s=o+jΩ Then for s = $$\frac{2}{T}(\frac{1-z^{-1}}{1+z^{-1}})$$, the values of Ω, ℴ are

A. ℴ = $$\frac{2}{T}(\frac{r^2-1}{1+r^2+2rcosω})$$, Ω = $$\frac{2}{T}(\frac{2rsinω}{1+r^2+2rcosω})$$

B. Ω = $$\frac{2}{T}(\frac{r^2-1}{1+r^2+2rcosω})$$, ℴ = $$\frac{2}{T}(\frac{2rsinω}{1+r^2+2rcosω})$$

C. Ω=0, ℴ=0

D. None

s = $$\frac{2}{T}(\frac{z-1}{z+1})$$

= $$\frac{2}{T}(\frac{re^jw-1}{re^jw+1})$$

= $$\frac{2}{T}(\frac{r^2-1}{1+r^2+2rcosω}+j \frac{2rsinω}{1+r^2+2rcosω})(s = ℴ+jΩ)$$

10. In equation ℴ = $$\frac{2}{T}(\frac{r^2-1}{1+r^2+2rcosω})$$ if r < 1 then ℴ < 0 and then mapping from s-plane to z-plane occurs in which of the following order?

A. LHP in s-plane maps into the inside of the unit circle in the z-plane

B. RHP in s-plane maps into the outside of the unit circle in the z-plane

C. All of the mentioned

D. None of the mentioned

In the above equation, if we substitute the values of r, ℴ then we get mapping in the required way

In equation ℴ = $$\frac{2}{T}(\frac{r^2-1}{1+r^2+2rcosω})$$ if r < 1 then ℴ < 0 and then mapping from s-plane to z-plane occurs in which LHP in s-plane maps into the inside of the unit circle in the z-plane.

11. In equation ℴ = $$\frac{2}{T}(\frac{r^2-1}{1+r^2+2rcosω})$$, if r > 1 then ℴ > 0 and then mapping from s-plane to z-plane occurs in which of the following order?

A. LHP in s-plane maps into the inside of the unit circle in the z-plane
B. RHP in s-plane maps into the outside of the unit circle in the z-plane
C. All of the mentioned
D. None of the mentioned

In equation ℴ = $$\frac{2}{T}(\frac{r^2-1}{1+r^2+2rcosω})$$, if r > 1 then ℴ > 0 and then mapping from s-plane to z-plane occurs in which RHP in s-plane maps into the outside of the unit circle in the z-plane

In the above equation, if we substitute the values of r, ℴ then we get mapping in the required way.

11. The general linear constant coefficient difference equation characterizing an LTI discrete time system is?

A. y(n)=-$$\sum_{k=1}^N a_k y(n-k)+\sum_{k=0}^N b_k x(n-k)$$

B. y(n)=-$$\sum_{k=0}^N a_k y(n-k)+\sum_{k=0}^N b_k x(n-k)$$

C. y(n)=-$$\sum_{k=1}^N a_k y(n)+\sum_{k=0}^N b_k x(n)$$

D. None of the mentioned

We know that, the general linear constant coefficient difference equation characterizing an LTI discrete time system is given by the expression

y(n)=-$$\sum_{k=1}^N a_k y(n-k)+\sum_{k=0}^N b_k x(n-k)$$

12. Which of the following is the rational system function of an LTI system characterized by the difference equation
y(n)=-$$\sum_{k=1}^N a_k y(n-k)+\sum_{k=0}^N b_k x(n-k)$$?

A. $$\frac{\sum_{k=0}^N b_k x(n-k)}{1+\sum_{k=0}^N a_k y(n-k)}$$

B. $$\frac{1+\sum_{k=1}^N a_k y(n-k)}{\sum_{k=0}^N b_k x(n-k)}$$

C. $$\frac{\sum_{k=0}^N b_k x(n-k)}{1+\sum_{k=1}^N a_k y(n-k)}$$

D. $$\frac{1+\sum_{k=0}^N a_k y(n-k)}{\sum_{k=0}^N b_k x(n-k)}$$

The difference equation of the LTI system is given as

y(n)=-$$\sum_{k=1}^N a_k y(n-k)+\sum_{k=0}^N b_k x(n-k)$$

By applying the z-transform on both sides of the above equation and by rearranging the obtained equation, we get the rational system function as

H(z)=$$\frac{\sum_{k=0}^N b_k x(n-k)}{1+\sum_{k=1}^N a_k y(n-k)}$$

13. We can view y(n)=-$$\sum_{k=1}^N a_k y(n-k)+\sum_{k=0}^N b_k x(n-k)$$ as the computational procedure (an algorithm) for determining the output sequence y(n) of the system from the input sequence x(n).

A. True
B. False

The computations in the given equation can be arranged into equivalent sets of difference equations. Each set of equations defines a computational procedure or an algorithm for implementing the system.

14. Which of the following is used in the realization of a system?

A. Delay elements
B. Multipliers
D. All of the mentioned

From each set of equations, we can construct a block diagram consisting of an interconnection of delay elements, multipliers, and adders.

15. Computational complexity refers to the number of ____________

B. Arithmetic operations
C. Multiplications
D. None of the mentioned

Computational complexity is one of the factors which is used in the implementation of the system. It refers to the numbers of Arithmetic operations (Additions, multiplications, and divisions).

16. The number of times a fetch from memory is performed per output sample is one of the factors used in the implementation of the system.
A. True
B. False

According to the recent developments in the design and fabrication of rather sophisticated programmable DSPs, other factors, such as the number of times a fetch from memory is performed or the number of times a comparison between two numbers is performed per output sample, have become important in assessing the computational complexity of a given realization of a system.

17. Which of the following refers to the number of memory locations required to store the system parameters, past inputs, past outputs, and any intermediate computed values?

A. Computational complexity
B. Finite world length effect
C. Memory requirements
D. None of the mentioned

Memory requirements refer to the number of memory locations required to store the system parameters, past inputs, past outputs, and any intermediate computed values.

18. Finite word length effects refer to the quantization effects that are inherent in any digital implementation of the system, either in hardware or software.

A. True
B. False

The parameters of the system must necessarily be represented with finite precision. The computations that are performed in the process of computing output from the system must be rounded off or truncated to fit within the limited precision constraints of the computer or hardware used in the implementation. Thus, Finite word length effects refer to the quantization effects that are inherent in any digital implementation of the system, either in hardware or software.

19. Which of the following are called finite word length effects?

A. Parameters of the system must be represented with finite precision
B. Computations are truncated to fit in the limited precision constraints
C. Whether the computations are performed in fixed-point or floating-point arithmetic
D. All of the mentioned

All three of the considerations given above are called finite word length effects.

20. The factors Computational complexity, memory requirements, and finite word length effects are the ONLY factors influencing our choice of the realization of the system.

A. True
B. False

Apart from the three factors given in the question, other factors such as, whether the structure or the realization lends itself to parallel processing or whether the computations can be pipelined are also the factors that influence our choice of the realization of the system.

21. 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.

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

23. 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.

24. 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.

25. 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.

26. 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.

27. 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%.

28. 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.

29. 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.

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

H(z)=$$\frac{1}{M}(1-z^{-M} e^{j2πα})\sum_{k=0}^{M-1}\frac{H(k+α)}{1-e^{j2π(k+α)/M} z^{-1}}$$
H1(z)=$$\frac{1}{M}(1-z^{-M} e^{j2πα})$$.