FIR Filter Design & Realizations of Digital Filters Electrical Engineering (EE) Notes | EduRev

Digital Signal Processing

Electrical Engineering (EE) : FIR Filter Design & Realizations of Digital Filters Electrical Engineering (EE) Notes | EduRev

The document FIR Filter Design & Realizations of Digital Filters Electrical Engineering (EE) Notes | EduRev is a part of the Electrical Engineering (EE) Course Digital Signal Processing.
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FIR filter design

In the previous section, digital filters were designed to give a desired frequency response magnitude without regard to the phase response. In many cases a linear phase characteristics is required through the passband of the filter. It can be shown that causal IIR filter cannot produce a linear phase characteristics and only special forms of causal FIR filters can give linear phase.

If {h[n]} represents the impulse response of a discrete time linear system a necessary and sufficient condition for linear phase is that {h[n]} have finite duration , that it be symmetric about its mid point, i.e

FIR Filter Design & Realizations of Digital Filters Electrical Engineering (EE) Notes | EduRev

For even, we get

FIR Filter Design & Realizations of Digital Filters Electrical Engineering (EE) Notes | EduRev

For N odd

FIR Filter Design & Realizations of Digital Filters Electrical Engineering (EE) Notes | EduRev

For even we get a non-integer delay, which will cause the value of the sequenceto change, [See continuous time implementation of discrete time system, for interpretation of non-integer delay].

One approach to design FIR filters with linear phase is to use windowing.

The easiest way to obtain an FIR filter is to simply truncate the impulse response of an IIR filter. If {hd[n]} is the impulse response of the designed FIR filter, then an FIR filter with impulseresponse {h[n]} can be obtained as follows.

FIR Filter Design & Realizations of Digital Filters Electrical Engineering (EE) Notes | EduRev

This can be thought of as being formed by a product of {hd[n]} and a window function {ω[n]}

FIR Filter Design & Realizations of Digital Filters Electrical Engineering (EE) Notes | EduRev

where {ω[n]}  is said to be rectangular window and is given by

FIR Filter Design & Realizations of Digital Filters Electrical Engineering (EE) Notes | EduRev

Using modulation property of Fourier transfer

FIR Filter Design & Realizations of Digital Filters Electrical Engineering (EE) Notes | EduRev

For example if  Hd(e) is ideal low pass filter and {ω[n]} is rectangular window is measured version of the ideal low pass frequency response.

FIR Filter Design & Realizations of Digital Filters Electrical Engineering (EE) Notes | EduRevFIR Filter Design & Realizations of Digital Filters Electrical Engineering (EE) Notes | EduRevFIR Filter Design & Realizations of Digital Filters Electrical Engineering (EE) Notes | EduRev

In general, the index the main lobe of  W(e), the more H(e) spreading where as the narrower the main lobe (larger N), the closer |H(e)| comes to. In general, we are left with a trade-off of making N large-enough so that smearing is minimized, yet small enough to allow reasonable implementation. Much work has been done on adjusting {ω[n]} to satisfy certain main lobe and side lobe requirements. Some of the commonly used windows are give in below. 

(a) Rectangular

FIR Filter Design & Realizations of Digital Filters Electrical Engineering (EE) Notes | EduRev

(b) Bartlett (or triangle)

FIR Filter Design & Realizations of Digital Filters Electrical Engineering (EE) Notes | EduRev

(c) Hanning

FIR Filter Design & Realizations of Digital Filters Electrical Engineering (EE) Notes | EduRev

(d) Harming

FIR Filter Design & Realizations of Digital Filters Electrical Engineering (EE) Notes | EduRev

(e) Blackman

FIR Filter Design & Realizations of Digital Filters Electrical Engineering (EE) Notes | EduRev

(f) Kaiser

FIR Filter Design & Realizations of Digital Filters Electrical Engineering (EE) Notes | EduRev

where  I0(x) is modified zero-order Bessel function of the first kind given by

FIR Filter Design & Realizations of Digital Filters Electrical Engineering (EE) Notes | EduRev

The main lobe width and first side lobe attenuation increase as we proceed down the window listed above.

An ideal lowpass filter with linear phase and cut off ωc is characterized by

FIR Filter Design & Realizations of Digital Filters Electrical Engineering (EE) Notes | EduRev

The corresponding impulse response is

FIR Filter Design & Realizations of Digital Filters Electrical Engineering (EE) Notes | EduRev

Since this is symmetric about n = α , if we change α = (N - 1)/2 and use one of the windows listed above the will get linear phase FIR filter. Transition width and minimum stopped attenuation are listed in the Table 9.3.

WindowTransition WidthMinimum stopband attenuation
Rectangular4π/N-21db
Bartlett8π/N-25dB
Hanning8π/N-44dB
Hamming8π/N-53dB
Blackman
 
12π/N-74dB
Kaiservariablevariable

Table 9.3

We first choose a window that satisfies the minimum attenuation. The transition bandwidth is approximately that allows us to choose the value of N. Actual frequency response characteristic are then calculated and we see if the requirements are met or not. Accordingly N is adjusted parameters for kaiser window are obtained from design formula available for this MATLAB or similar programmes have all there formulas.

Realizations of Digital Filters

We have many realizations of digital filter. Some of these are now discussed. Direct Form Realization - An important class of linear time -invariant systems is characterized by the transfer function.

FIR Filter Design & Realizations of Digital Filters Electrical Engineering (EE) Notes | EduRev

A system with input {x[n]} and output {y[n]} could be realized by the following constant coefficient difference equation

FIR Filter Design & Realizations of Digital Filters Electrical Engineering (EE) Notes | EduRev

A realization of the filter using equation (9.31) is shown in figure (9.6)

FIR Filter Design & Realizations of Digital Filters Electrical Engineering (EE) Notes | EduRev

The output y[n] is seen to be weighted sum of input x[n] and past inputs x[n - 1],.....x[n - A] and past outputs. Another realization can be obtained by uniting H(z) as product of two transfer functions H1(z) and H2(z), where H1(z) contains only the denominator or poles and H2(z) contains only the numerator or zeros as follows

FIR Filter Design & Realizations of Digital Filters Electrical Engineering (EE) Notes | EduRev

where

FIR Filter Design & Realizations of Digital Filters Electrical Engineering (EE) Notes | EduRev

The output of the filter is obtained by calculating the intermediate result {ω[n]} obtained from operating on the input with filter H1(z) and then operating on w[n] with filter.Thus we obtain 

FIR Filter Design & Realizations of Digital Filters Electrical Engineering (EE) Notes | EduRev

or

FIR Filter Design & Realizations of Digital Filters Electrical Engineering (EE) Notes | EduRev

and

FIR Filter Design & Realizations of Digital Filters Electrical Engineering (EE) Notes | EduRev

or

FIR Filter Design & Realizations of Digital Filters Electrical Engineering (EE) Notes | EduRev

The realization is shown in figure 9.8

FIR Filter Design & Realizations of Digital Filters Electrical Engineering (EE) Notes | EduRev

Upon close examination of Fig 9.8, it can be seen that the two branches of delay elements can be combined as they both refer to delayed versions of ω[n] and upon simplification, the direct form II canonical realization is obtained as shown in figure 9.9

FIR Filter Design & Realizations of Digital Filters Electrical Engineering (EE) Notes | EduRev

In this form the number of delay element is max (M,N). It can be shown that this is the minimum number of delay elements that are required to implement the digital filter. This does not mean that this is the best realization. Immunity to roundoff and quantization are very important considerations.

An important special case that is used as building block occurs when. Thus H(z)  is ratio of two qualities in z-1, called biquadratic section, and is given by

FIR Filter Design & Realizations of Digital Filters Electrical Engineering (EE) Notes | EduRev

The alternative form is found to be useful for amplitude scaling for improving performance file filter operation. This form is shown in figure 9.10.

FIR Filter Design & Realizations of Digital Filters Electrical Engineering (EE) Notes | EduRev

Cascade RealizationsIn the cascade realization H(z) is broken into productof transfer functions  FIR Filter Design & Realizations of Digital Filters Electrical Engineering (EE) Notes | EduRev  each a rational expression in z-1 as follows

FIR Filter Design & Realizations of Digital Filters Electrical Engineering (EE) Notes | EduRev

FIR Filter Design & Realizations of Digital Filters Electrical Engineering (EE) Notes | EduRev

H(z)  could be broken up in many ways; however the most common method is to use biquadratic sections. Thus

FIR Filter Design & Realizations of Digital Filters Electrical Engineering (EE) Notes | EduRev

by letting b2k and a2k  equal to zero we get bilinear section. Even among the biquadratic sections we have many choices as how we pair poles and zeros. Also the order of the sections can be different

Example:

Final cascade realization of

FIR Filter Design & Realizations of Digital Filters Electrical Engineering (EE) Notes | EduRev

Using only real coefficients H(z)  can be decompressed as

FIR Filter Design & Realizations of Digital Filters Electrical Engineering (EE) Notes | EduRev

Divides both numerator and denominator by z3 and factoring 8 as  FIR Filter Design & Realizations of Digital Filters Electrical Engineering (EE) Notes | EduRev , one possible rearrangement for H(z) is

FIR Filter Design & Realizations of Digital Filters Electrical Engineering (EE) Notes | EduRev

This can be realized as shown is figure 9.12

FIR Filter Design & Realizations of Digital Filters Electrical Engineering (EE) Notes | EduRev

Parallel Realizations:

The transfer function (z) could be written as a sum of transfer functionsFIR Filter Design & Realizations of Digital Filters Electrical Engineering (EE) Notes | EduRev as follows:

FIR Filter Design & Realizations of Digital Filters Electrical Engineering (EE) Notes | EduRev

One parallel form results when Hk(z) are all selected to be of the following form for (M < N) 

FIR Filter Design & Realizations of Digital Filters Electrical Engineering (EE) Notes | EduRev

If M > N, we will have a section H0(z) of FIR filter, obtained by performing long division. Once denominator polynomial has degree more than the numerator polynomial we perform the partial fraction expansion. The resulting structure is shown in figure 9.13.

FIR Filter Design & Realizations of Digital Filters Electrical Engineering (EE) Notes | EduRev

Example:

Find the parallel form for the filter given in last example.

FIR Filter Design & Realizations of Digital Filters Electrical Engineering (EE) Notes | EduRev

Using MATLAB program or otherwise we get

FIR Filter Design & Realizations of Digital Filters Electrical Engineering (EE) Notes | EduRev

using direst form realization for individual section we get the structure shown in figure 9.14.

FIR Filter Design & Realizations of Digital Filters Electrical Engineering (EE) Notes | EduRev

Apart from these there exist a number of other realizations like lattice form, state variable realization etc.  

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