FIR Filters Design Comparison MCQ [Free PDF] – Objective Question Answer for FIR Filters Design Comparison Quiz

Digital filters are the filters that can be designed from analog filters which have infinite duration unit sample response.

There are many techniques that are used to convert analog filters into digital filters of which some of which are Approximation of derivatives, bilinear transformation, impulse invariance, and many other methods.

An FIR filter of length M with input x(n) and output y(n) is described by the difference equation

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

where {bk} is the set of filter coefficients.

We know that the impulse response h(t) and the Laplace transform Ha(s) are related by the equation.

Ha(s)= \( \int_{-\infty}^{\infty} h(t)e^{-st} dt\)

There are many ways how we represent a system function of which one is normal representation i.e., output/input and other ways like Laplace transform and rational system function.

An analog linear time-invariant system with system function H(s) is stable if all its poles lie on the left half of the s-plane.

If the conversion technique is to be effective, the jΩ axis in the s-plane should map into the unit circle in the z-plane. Thus there will be a direct relationship between the two frequency variables in the two domains.

If the conversion technique is to be effective, then the LHP of the s-plane should be mapped into the inside of the unit circle in the z-plane. Thus a stable analog filter will be converted to a stable digital filter.

If an IIR filter is stable and if it can be physically realizable, then the filter cannot have a linear phase.

A linear phase filter must have a system function that satisfies the condition

H(z)=\(±z^{-N} H(z^{-1})\)

where z(-N) represents a delay of N units of time.

For a linear phase filter, we know that

H(z)=\(±z^{-N} H(z^{-1})\)

where z(-N) represents a delay of N units of time. But if this were the case, the filter would have a mirror image pole outside the unit circle for every pole inside the unit circle. Hence the filter would be unstable.

The order of operations to be performed in order to realize linear phase IIR filter

(i) Passing x(-n) through a digital filter H(z)
(ii) Time reversing the output of H(z)
(iii) Time reversal of the input signal x(n)
(iv) Passing the result through H(z)

If the restriction on physical reliability is removed, it is possible to obtain a linear phase IIR filter, at least in principle. This approach involves performing a time-reversal of the input signal x(n), passing x(-n) through a digital filter H(z), time reversing the output of H(z), and finally, passing the result through H(z) again.

The signal processing is computationally cumbersome and appears to offer no advantages over linear phase FIR filters. Consequently, when an application requires a linear phase, it should be an FIR filter.

One of the simplest methods for converting an analog filter into a digital filter is to approximate the differential equation by an equivalent difference equation.

For the derivative dy(t)/dt at time t=nT, we substitute the backward difference [y(nT)-y(nT-T)]/T. Thus
dy(t)/dt =[y(nT)-y(nT-T)]/T
=[y(n)-y(n-1)]/T

where T represents the sampling interval and y(n)=y(nT).

The analog differentiator with output dy(t)/dt has the system function H(s)=s, while the digital system that produces the output [y(n)-y(n-1)]/T has the system function H(z) =(1-z-1)/T. Thus the relation between s-domain and z-domain is given as
s=(1-z-1)/T.

We know that dy(t)/dt =[ y(n)-y(n-1)]/T

Second order derivative of y(t) is d(dy(t)/dt)/dt=[y(n)-2y(n-1)+y(n-2)]/T2.

We know that for a second order derivative

d2y(t)/dt²=[y(n)-2y(n-1)+y(n-2)]/T2

=>s2 = \(\frac{1-2z^{-1}+z^{-2}}{T^2} = (\frac{1-z^{-1}}{T})^2\)