IIR FILTER DESIGN
1.Give the expression for location of poles of normalized Butterworth filter
The poles of the Butterworth filter is given by (S-S1)(S- S2)……..(S-SN)
Where N is the order of filter.
2.What are the parameters(specifications) of a Chebyshev filter?
From the given chebyshev filter specifications we can obtain the parameters like the order of the filter N, ε, transition ratio k, and the poles of the filter.
3.Find the digital transfer function H(z) by using impulse invariant method for the analog transfer function H(s)=1/s+2.Assume T=0.5sec
H(z)= 1/ 1-e-1z-1
4.What is Warping effect?
The relation between the analog and digital frequencies in bilinear transformation is given by
For smaller values of ω there exist linear relationship between ω and Ω. But for large values of ω the relationship is non-linear. This non-linearity introduces distortion in the frequency axis. This is known as warping effect. This effect compresses the magnitude and phase response at high frequencies.
5.Compare Butterworth & Chebyshev filter.
1. All pole design
2. The poles lie on a circle in S-plane.
3. The magnitude response is maximally flat at the origin and monotonically decreasing function of Ω
4. The normalized magnitude response has a value of 1/√2 at the cutoff frequency Ωc
5. Only few parameters has to be calculated to determine the transfer function
1. All pole design
2. The poles lie on ellipse in S-plane.
3. The magnitude response is equiriple in passband and monotonically decreasing in the stopband.
4. The normalized magnitude response has a value of 1/√1+ε↑2 at the cutoff frequency Ωc.
5. A large number of parameter has to be calculated to determine the transfer function
If H(s)= Σ Ck / S-Pk then H(z) = Σ Ck / 1-ePkTZ-1
7. What is bilinear transformation?
The bilinear transformation is a mapping that transforms the left half of s plane into the unit circle in the z-plane only once, thus avoiding aliasing of frequency components. The mapping from the s- plane to the z-plane in bilinear transformation is s =
8.What is the main disadvantage of direct form-I realization?
The direct form realization is extremely sensitive to parameter quantization. When the order of the system N is large, a small change in a filter coefficient due to parameter quantization, results in a large change in the location of the pole and zeros of the system.
9.What is Prewarping?
The effect of the non-linear compression at high frequencies can be compensated. When the desired magnitude response is piece-wise constant over frequency, this compression can be compensated by introducing a suitable prescaling, or prewarping the critical frequencies by using the formula, Ω=2/T tan ω/2.
10.List the features that make an analog to digital mapping for IIR filter design coefficient.
The bilinear transformation provides one-to-one mapping.
Stable continuous systems can be mapped into realizable, stable digital systems.
There is no aliasing.
In impulse invariant method, the mapping from s-plane to z-plane is many to one i.e., all the poles in the s-plane between the intervals [(2k-1)π]/T to [(2k+1)π]/T ( for k=0,1,2……) map into the entire z-plane. Thus, there are an infinite number of poles that map to the same location in the z-plane, producing aliasing effect. Due to spectrum aliasing the impulse invariance method is inappropriate for designing high pass filters. That is why the impulse invariance method is not preferred in the design of IIR filter other than low pass filters.
12.Find digital transfer function using approximate derivative technique for the analog transfer function H(s)=1/s+3.Assume T=0.1sec
H(z) = 1/ Z+e-0.3
13.Give the square magnitude function of Butterworth filter.
The magnitude function of the butter worth filter is given by
Where N is the order of the filter and Ωc is the cutoff
frequency. The magnitude response of the butter worth filter closely approximates the ideal response as the order N increases. The phase response becomes more non-linear as N increases.
14. Find the digital transfer function H(z) by using impulse invariant method for the analog transfer function H(s)=1/s+1.Assume T=1sec.
H(z)= 1/ 1-e-1z-1
15.Give the equation for the order of N and cut-off frequency Ωc of butter worth filter.
16.What are the properties of the bilinear transformation?
The mapping for the bilinear transformation is a one-to-one mapping; that is for every point z, there is exactly one corresponding point s, and vice versa.
The jΩ-axis maps on to the unit circle |z|=1, the left half of the s-plane maps to the interior of the unit circle |z|=1 and the right half of the s-plane maps on to the exterior of the unit circle |z|=1.
17.Write a short note on pre-warping.
The effect of the non-linear compression at high frequencies can be compensated. When the desired magnitude response is piece-wise constant over frequency, this compression can be compensated by introducing a suitable pre-scaling, or pre-warping the critical frequencies by using the formula. Ω =
18.What are the different types of structure for realization of IIR systems?
The different types of structures for realization of IIR system are
Transposed direct-form II structure
Cascade form structure
Parallel form structure
19.Draw the general realization structure in direct-form I of IIR system.
20.Give direct form II structure.
21.Draw the parallel form structure of IIR filter.
22.Mention any two techniques for digitizing the transfer function of an analog filter.
The two techniques available for digitizing the analog filter transfer function are Impulse invariant transformation and Bilinear transformation.
23.Write a brief notes on the design of IIR filter. (Or how a digital IIR filter is designed?)
For designing a digital IIR filter, first an equivalent analog filter is designed using any one of the approximation technique for the given specifications. The result of the analog filter design will be an analog filter transfer function Ha(s). The analog filter transfer function is transformed to digital filter transfer function H(z) using either Bilinear or Impulse invariant transformation.
24.Define an IIR filter
The filters designed by considering all the infinite samples of impulse response are called IIR filers. The impulse response is obtained by taking inverse Fourier transform of ideal frequency response.
25. Compare IIR and FIR filters.
i. All the infinite samples of impulse response are considered.
ii. The impulse response cannot be directly converted to digital filter transfer function.
iii. The design involves design of analog filter and then transforming analog filter to digital filter.
iv. The specifications include the desired characteristics for magnitude response only.
v. Linear phase characteristics cannot be achieved.
i. Only N samples of impulse response are considered.
ii. The impulse response can be directly converted to digital filter transfer function.
iii. The digital filter can be directly designed to achieve the desired specifications.
iv. The specifications include the desired characteristics for both magnitude and phase response.
v. Linear phase filters can be easily designed.
Usually, in the IIR Filter design, Analog filter is designed, then it is transformed to a digital filter the conversion of Analog to Digital filter involves mapping of desired digital filter specifications into equivalent analog filter.
The analog Frequency is same as the digital frequency response. At high frequencies, the relation between ω and Ω becomes Non-Linear. The Noise is introduced in the Digital Filter as in the Analog Filter. Amplitude and Phase responses are affected by this warping effect.
The Warping Effect is eliminated by prewarping of the analog filter. The analog frequencies are prewarped and then applied to the transformation.
Infinite Impulse Response:
Infinite Impulse Response filters are a Type of Digital Filters which has infinite impulse response. This type of Filters are designed from analog filters. The Analog filters are then transformed to Digital Domain.
Bilinear Transformation Method:
In Bilinear transformation method the transform of filters from Analog to Digital is carried out in a way such that the Frequency transformation produces a Linear relationship between Analog and Digital Filters.
A filter is one which passes the required band of signals and stops the other unwanted band of frequencies.
The Band of frequencies which is passed through the filter is termed as passband.
The band of frequencies which are stopped are termed as stop band.
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