FIR Filter Design

Structure:

Figure: FIR Filter structure

Steps:

Find coefficients:

I used Scilab for the same. Though, the procedure for the same is explained in Rulph Chassiang’s DSP book. The equation is mentioned in the book. I just expressed the equation in Scilab to get M coefficients.

I followed the window's derivation method for finite response.

Above image depicts how to get the equation for the Fourier series.

v = normalized frequency = f / F (Nyq)

Figure: Transfer function in frequency domain for a) lowpass, b) highpass, c) bandpass, d) bandstop filters

To compute the Fourier series coefficients for the filters:

Figure: Transfer function for the filters

So, for a LP filter:

v (normalized freq) = 0.2, coefficient calculation is shown.

Mathematically, the idea is to convert the transfer function of the required filter in Time domain, which would be an infinite impulse response. So, we have to convolute the response with window function (I had used rectangular window).
And the derived signal could be convoluted in time domain with the base signal.

I derived the above, equation using Scilab and convoluted with Rectangular window function w(n):

Scilab script:

M =11

N =M-1;

U =N / 2;

h_Rect=window('re', M) 

//hd(n) -> time domain function of Low Pass Filter

for n = 1:1:N+1

if (n-1) ==U

 hd(n) =0.4;

else

  hd( n) =( sin(2* %pi *( (n-1) - U ) /5) ) /( %pi *( (n-1) - U) ) ;

end

  h( n ) = hd( n ) * h_Rect( n ) ;

end

H_f= fft(h)

H = abs(H_f)

plot(H)

//v (normalized freq) for Low Pass cut-off here is 0.4 (2/5)

Signal:

t = (0:0.5:10)

//signal with frequency peaks at 0.2 and 0.8 on normalized frequency scale

s_t = sin(2 * %pi * 0.2 * t) + sin(2 * %pi * 0.8 * t)

Filter output y(n) is calculated as:

M =11

s_t = {-1.225D-15, -1.1755705, -3.331D-16, -1.902113, 5.551D-16, -3.674D-16, 3.331D-16, 1.902113, -1.110D-16, 1.1755705, 1.00000004581D-18}

//latest sample is first. s_t(0) sample is the last element in the list

y = 0

for k = 1:1:M

y = y + (h(k) * s_t(k))

mat(1,k) = h(k) * s_t(k)

end

mat(1,12) = 0

for k=2:1:5

for m = 1:2:M

mat(k,(m+1)/2) = mat(k-1,m) + mat(k-1, m+1)

end

end

//y = output sample of that particular instance and mat represents the calculation of y at each step in rtl design.

RTL Design:

Once design is ready. You can start implementing the design using Hardware Description Language.

I cannot post the entire RTL code I designed for the same here, though I will post some hints here:

1. Study the IEEE Floating Point Number representation format.

2. Understand the basic arithmetic operations in Floating Point numbers.

3. Don't create unnecessary module hierarchies. Use functions/tasks to implement Floating Point arithmetic.

4. In the figure which represents the structure of the filter, visualize a flop or memory where a block with 'z' is placed. Every 'z' represents the sampled value at previous instance. Convolution is implemented over a time period.

Implementation:

After, RTL Design is ready, take the design through implementation flows including Synthesis, Scan Insertion and Physical Design.