Designing IIR Filters

Designing filters involves the

mapping and transformation of well-known continuous time filter designs to discrete time ones

Impulse Invariance means that the

Impulse response of the digital filter will be similar to that of the continuous time filter

How can the impuulse response of a CT Filter be similar to that of a DT Filter or Digital filters

by sampling of the impulse response of the CT Filter

Impulse invariance maps the _ to -

S domain to the Z domain

Design Procedure of Digital filters when using Impulse invariance involves

* Starting with an analogue prototype or Transfer Function H(s)
* Find the inverse Laplace Transform to obtain analogue prototype impulse response h(t)
* Sample the impulse response h\[n\] = h\[nT\]
* The desired transfer function is found by taking the z transform of h\[n\]

Result of Sampling the analogue prototype impulse response gives

* an impulse response of the discrete filter h\[n\] that is identical to the analogue filter at t = nT
* sampling in the time domain will cause frequency aliasing

Impulse response transformation involves

the conversion of t→ n = nT

Frequency Response Transformation involves

the conversion of ώ → Ω = ώT

Design Limitations of the Impulse Invariance is

* The frequency response of the CT filter has to be bandlimited for aliasing not to occur
* High pass and band -stop filter design is not possible with impulse invariance
* Practical continuous time filters are not exactly band limited and some aliasing occurs
* In practise to compensate for aliasing the continuous time filter may be overdesigned ( lower cut- off, higher attenuation in stop-band)

Impulse invariance indirectly maps the

s-plane to the z-plane through sampling of the impulse response

Bilinear Transformation

applies a non-linear mapping of the analogue frequency axis to the digital frequency one

IIR filter design steps involve

* specification of the continuous -time prototype low-pass filter
* frequency transformation of the analogue prototype low-pass filter
* transformation of the CT filter to DT filter

Types of Analogue prototype filters

* Butterworth Analogue filter
* Chebychev Analogue Filter
* Elliptic analogue filter

Butterworth Analogue Filter

* Maximal flatness at Ω = 0 and Ω = ∞
* Maximum distortion at Ω = Ω_c
* Not a very sharp cut-off

Chebychev filter specification

* Equiripple passband (type I)
* Equiripple stopband (type II)
* sharper cut off compared to butterworth
* lower order filter needed to satisfy the requirements

if the chebychev and butterwort filter have the same design specification for passband and stop band edge , pass band ripple and minimum attenuation, Who has a better performance

Chebychev because it has a sharper transition band

non linear mapping involves

−∞≤ώ≤∞ → −π≤ώ≤π

properties of bilinear transformation

* stable CT filter maps to a stable DT filter i.e half of s plane maps to inside the unit circle
* the *j*ώ axis maps on the unit circle

s and z relationship is given as

s = 2/T\[(1-z¹)/(1-z¹)\]

where z¹ = z^-1

ώ relationship to Ω

ώ = 2/T tan(Ω/2)

Frequency Warping

non-linear compression of the frequency axis

to solve the problem of frequency warping or to compensate the warping effect

pre-warp the frequency design specifications of the DTfilter using

ώ=2tan(Ω/2)/T

Steps in using pre-warping to get desired analogue frequency

* The desired stop- band frequency is given
* Pre-warp to get analogue filters requires stop band frequency
* bilinear transformation is applied s → z
* results in a digital filter with the desired stop-band

Design Procedure using Bilinear Transformation

* Incorporate frequency axis pre-warping as appropriate
* select analogue filter prototype H(s) to satisfy design spec (Butterworth, Chebychev)
* Apply Frequency Transformation ( changing from lowpass to any type)
* Evaluate H(s) and apply bilinear transform to obtain H(z)

Bilinear Transformation Disadvantages

* Frequency axis warping
* Distortion of phase characteristics
* Distortion of time domain characteristics

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Direct Optimisation

* Specify the design objectives
* Choose filter structure and paramters
* Select optimisation tool to find the optimal or sub optimal set of filter parameters