FIR Filter Design

 

Filter Design by window method - Rectangular, Haanning and Hamming Window

Filter

- Filter is a system that is used to modify certain frequencies relative to others.
- It passes certain frequency components and totally rejects all other.

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Linear Phase FIR Filter

- The system function of an FIR filter of length N is given by:
H(z) = ∑n=0to N-1 { h[n] z-n }

- The corresponding frequency response is given by substituting the value of z with ejΩ :
H(ejΩ) = ∑n=0to N-1 { h[n] ejΩn } = H(Ω) . ejϕ(Ω)
where,
H(Ω) is zero phase frequency response
ϕ(Ω) is a phase term

- A filter is called linear phase filter if the phase term has the form of:
ϕ(Ω) = αΩ + β

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Types of linear phase FIR filter

- A filter will have linear phase if its impulse response is symmetric or anti symmetric.
h[N-1-n] = (+ or -) h[n] for all n

- There are four types of linear phase FIR filters. They are as follows:
1. N odd and h[n] symmetrical (Type I)
2. N even and h[n] symmetrical (Type II)
3. N odd and h[n] asymmetrical (Type III)
4. N even and h[n] asymmetrical (Type IV)

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FIR Filter Design using Windows

- FIR filter is designed by window method by simply multiplying the infinitely long impulse response by a window function.

- Let us consider the desired frequency of the digital filter be Hd(ejΩ).
So, the impulse response of the required filter is given by:
hd[n] = 1/2π . ∫-π to π {Hd(ejΩ) . (ejΩn) . dΩ}

- The impulse response obtained by the digital filter may not be of finite duration.

- An approximate FIR filter of finite duration can be obtained by truncating the impulse response. This truncation can be done with the help of a window function such that:
h[n] = hd[n] . w[n]
where,
w[n] is a window function.

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Rectangular Window

- Rectangular window is a window function having magnitude 1 for the range of n = 0 to M-1.
- It is denoted by wR[n].

- Mathematically,

wR[n] = 1 for 0 <= n <= M-1
                                  = 0 elsewhere

 

 

Let us consider the fourier transform of rectangular window as:
     WR(w) = ∑n = 0 to M-1 { wR(n) . e-jwn }
                                         = ∑n = 0 to M-1 { e-jwn }

 

From the standard series relation:
WR(w)
= [e-jw*0 - e-jwM ] / [ 1 - a ]
= [ 1- e-jwM ] / [ 1 - e-jw ]
= [ {e-jwM/2 . ejwM/2} - {e-jwM/2 . e-jwM/2} ] / [ {e-jw/2 . ejw/2} - {e-jw/2 . e-jw/2} ]
= [ e-jwM/2 {ejwM/2 - e-jwM/2} ] / [ e-jw/2 {ejw/2 - e-jw/2} ]
= [ e-jw{(M-1)/2} ] * [ sin(wM/2) ] / [ sin(w/2) ]

Comparing with polar form :
The magnitude response of rectangular window is given by:
| WR(w) | = | [ sin(wM/2) ] / [ sin(w/2) ] |

 

The magnitude response for M = 11 is shown in figure below:

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Problems with Rectangular Window:

- With increase in window length, width of main lobe in frequency response decreases while side lobes are unaffected.
- Large side lobes in W(Ω) cause undesirable ringing in the filter designed.

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Hamming Window

- Hamming window is given by:

      wH(n) = 0.54 - 0.46 cos (2πn / (M-1)) for n = 0, 1, 2, .......... , M-1
                                          = 0 ; elsewhere

 

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Hanning Window

- Hanning window is given by:

      wHN(n) = 1/2 [ 1 - cos (2πn / (M-1)) ]   for n = 0, 1, 2, .........., M-1 
                                          = 0 ; elsewhere

 

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Filter Design by Kaiser Window

- In other windows, width of main lobe and attenuation of side lobes depend only upon window length M.
- Kaiser window allows separate control of these factors.

- Kaiser window is defined as:

     wk(n) = [ Io { β [ 1 - ((n-α)/α)2]1/2} ] / [ Io(β) ]      for 0 <= n <= M
                    = 0 ; elsewhere

 

where, Io() is 0th order modified Bessel function of first kind.
α = M/2
β is called shape parameter

 

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Design Steps

1. Get desired frequency response Hd(w)
2. Take inverse fourier transform of Hd(w) to obtain hd[n]
3. Decide length of FIR filter
4. Multiply hd[n] by selected window function to get h[n].

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Q) Design linear phase filter using Kaiser window to meet the given specification.

 

The given specification is:
            0.99 <= | H(ejw | <= 1.01 ; for 0 >= w >= 0.19π
            | H(ejw | <= 0.01 ; for 0.21π <= w <= π

 

From the above specification, we get:
δp = 0.01
δs = 0.01
wp = 0.19π
ws = 0.21π

Now,
Δw = ws - wp = 0.02π
Minimum value of ripple (δ) = min(δp, δs) = 0.01

Attenuation in dB is:
A = -20 log10δ = 40 dB

Cut off frequency is:
wc = { ws + wp } / 2 = 0.2π

Now,
As, 21 <= A <= 50; the value of β is given by:
β = 0.5842 (A-21)0.4 + 0.07886 (A-21) = 3.395

Window width is calculated as:
M = (A-8) / 2.285Δw = 222.88 = 223

The value of α is given by:
α = M / 2 = 111.5

Now, the Kaiser window is defined by:
wk(n) = [ Io { β [ 1 - ((n-α)/α)2]1/2} ] / [ Io(β) ] for 0 <= n <= M
= 0 ; elsewhere

So,
wk(n) = [ Io { 3.395 [ 1 - ((n-111.5)/111.5)2]1/2} ] / [ Io(3.395) ] for 0 <= n <= M
= 0 ; elsewhere

Now,
The low pass filter has desired frequency response of :
Hd(w) = e-jw((M-1)/2) ; for -wc <= w <= wc
= 0 ; elsewhere

Taking inverse fourier transform:
hd(w) = sin[wc (n - (M-1)/2)] / [ π (n - (M-1)/2) ] ; for n != (M-1) / 2
= wc / π ; for n = (M-1) / 2

As the desired filter length is M+1;
hd(w) = sin[wc (n - M/2)] / [ π (n - M/2) ] ; for n != M / 2
= wc / π ; for n = M / 2

As M/2 is not a integer;
hd(w) = sin[wc (n - M/2)] / [ π (n - M/2) ]
= sin[0.02π (n - 111.5)] / [ π (n - 111.5) ] ; for 0 <= n <= M

Hence, the required filter design is obtained by:
h[n] = hd(n) . wn

 

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Q) Design FIR filter having pass band edge frequency w_p = 0.03π, w_s = 0.05π and α_s = 40 dB using appropriate window function.

 

Here, αs = 40 dB which is close to 44 dB (Hanning window). So, we choose Hanning window for the design of the given FIR filter.

 

Given:
wp = 0.03π
ws = 0.05π

For Hanning window, width of main lobe = 8π / M = k (2π / M )
So, k = 4

And,
M = k . [ 2π / { ws - wp } ] = 400

The Hanning window function is given by:
wHN(n) = 1/2 [ 1 - cos (2πn / 399) ] for n = 0, 1, 2, .........., M-1
= 0 ; elsewhere

The desired frequency response of LPF is given by:
Hd(w) = e-jw((M-1)/2) ; for -wc <= w <= wc
= 0 ; elsewhere

Taking the inverse fourier transform:
hd(w) = sin[wc (n - (M-1)/2)] / [ π (n - (M-1)/2) ] ; for n != (M-1) / 2
= wc / π ; for n = (M-1) / 2

Hence, the required filter design is obtained by:
h[n] = hd(n) . wHN(n)

 

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Filter Design by Frequency Sampling Method

- The desired frequency response is specified as samples at a set of equally spaced frequencies.
- h(n) = 1/N ∑n = 0 to M-1 { H(k + α) . e-j 2π(k+α) n/M } ; n = 0, 1, ....., M-1

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Filter Design by Optimum Approximation

- Optimal filter is a filter that is designed to spread the weighted approximation error between the desired frequency response and the actual frequency response across the passband and stopband of the filter.
- The main aim is to reduce the maximum error.
- The result has ripples in both passband and stopband.

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Remez Exchange Algorithm

Definition:

- Remez exchange algorithm is an iterative algorithm to determine the set of filter parameters as required by alternation theorem.

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Flowchart:

- The flow chart for Remez Exchange Algorithm is shown in given figure:

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Derivation:

The real valued frequency response can be written as:

          Hr(Ω) = Q(Ω) . P(Ω)
where,
  P(Ω) = ∑k = 0 to L {α(k) . cos Ωk}

 

Q(Ω) = 1 ; case 1
= cos Ω/2 ; case 2
= sin Ω ; case 3
= sin Ω/2 ; case 4

αk are filter parameters.

 

Let the desired real valued frequency response be Hdr(Ω).

The error between desired and designed frequency response is controlled with weighted error function as:
E(Ω) = W(Ω) [ Hdr(Ω) - Hr(Ω) ]

The weight function is normalized as:

    W(Ω) = δ2 / δ1 ; Ω in pass band
              = 1 ; Ω in stop band

 

Now,
E(Ω) = W(Ω) Q(Ω) [ Hdr(Ω) / Q(Ω) - P(Ω) ]

 

 

Let W'(Ω) = W(Ω) . Q(Ω)
       H'dr(Ω) = Hdr(Ω) / Q(Ω)

 

So,
E(Ω) = W'(Ω) [ H'dr(Ω) - P(Ω) ]

 

Now,
The filter parameters should be obtained such that:

minover {α(k)} [ maxw ∈ S { |E(Ω)| }] = minover {α(k)} [ maxw ∈ S { |W'(Ω) [ H'dr(Ω) - P(Ω) ]| }]