FIR is a class of LTI filter and the transfer function of the filter is

Now consider an FIR filter having length M can be expressed as

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=  =

We obtained the impulse response with

The impulse response of finite length   M . FIR
filter have only Zeros .FIR filter is also known as all zeros filter or feed
forward or  non -recursive  or transversal.

For  linear phase response  FIR digital
filter satisfies the following condition

we have to
know that

Inverse
Discrete time Fourier transform

For high
pass filter limit breakdown into two interval

since  sin(n)=0
so above equation become

Where
and

Impulse
response shows non-causal behaviour  and to
convert it  into causal system by giving
a  delay . Delaying term convert into
linear phase and causal .

for odd

The impulse
response truncation is equal to multiplying the

by the
selecting  window. In given case the stop
band attenuation is 74db which is near to the Blackman window.

Blackman
window mathematical expression can be expressed as

for

The final
expression for  is

Designing parameters of  high pass filter:

Stopband : 0-2.5 kHz

Passband : 3
kHz

Sampling
rate: 10 kHz

Stopband
attenuation:72 dB

Sol:

According
to the stopband attenuation -72dB we select the Blackman window.

for

For   M=110

for

The final
output can be expressed as

n

0

-0.00819

-0.08

0.000652

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109

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Matlab
Code:

>> L=110;

>> Xc=0.55;

>> Qn=blackman(L+1);

>> p=fir1(L,Xc,’high’ ,Qn);

>> freqz(p)

Outp