A filter is an electrical network that transmits signals within a specified frequency range called the pass band, and suppresses signals in the stop band, separated by the cut-off frequency. Digital filters are used to eliminate noise and extract signals of interest, implemented using software rather than RLC components. Digital filters are FIR (finite impulse response) or IIR (infinite impulse response) depending on the number of sample points used. An ideal filter would transmit signals in the pass band without attenuation and completely suppress the stop band, but ideal filters cannot be realized. IIR filter design first develops an analog IIR filter, then converts it to digital using methods like impulse invariant, approximation of derivatives, or bilinear transformation.

1.  What is Filter ?
A Filter is an electrical network that can transmit signal
within a specified frequency range .
This Frequency range is called PASS BAND and
Where signal is suppressed is called STOP BAND.
The Frequency that separates the pass band and the stop
band is known as CUT-OFF frequency.
 What is Digital filter and why we use it ?
Type of filter –
1) Analog filter 2) Digital filter
Digital filter use to eliminate the Noise and to Extract the
signal of interest from the other signal .
Filtering is not done by RLC it done by using Difference eqn.
It is implement using software like C and Assemble .

2.  Types of Digital Filter –
Depending on the no. of sample point used to
determine the unit sample (impulse response)of LTI
system ,digital filter are 2 types –
1)FIR(finite impulse response)filter
2)IIR(infinite impulse response)filter
 What is Ideal Filter?
An ideal filter is would transmit signal under the pass
band without attenuation and completely suppress the
signal in stop band.
Characteristics –
 it have constant gain in pass band and zero gain in the
stop band.
 It has linear phase response.
 It must be causal .

3.  Ideal filter can not be realize .
IIR Filter Design –
first design analog IIR filter. Then analog filter converted into
the digital filter .
Methods –
1) Impulse Invariant method – In this, we match the analog filter
impulse response to the digital response.
2) Approximation of derivative method – In this ,differential
equation of analog filter transform into difference equation of digital
filter.
3) Bilinear method— In this, conformal mapping is done which
transform the jΩ-Axis into the unit circle in the Z-plane.

4.  What is basic analog filter Approximation?
Approximation of analog filter is required because the practical
characteristic of a filter is not identical to ideal characteristic .
The approximation are 3 types -
1)Butterworth filter approximation
2) Chebyshev filter approximation
3) Elliptic filter approximation
Note- The approximation is used for achieve ideal
characteristic where as the methods are use to
transform the analog filter response to digital filter
response. we use transformation methods for
designing of ideal characteristic.

5. Butterworth Filter
 For low pass filter.

6.  The main characteristic of Butterworth filter is that the
pass band is maximally flat. There are no variation
(ripples) in the pass band.
 The magnitude response of LP Butterworth filter
is given by -
Where
Ωc =cut off frequency
|H(Ω)|2 =magnitude of LPF
N= order of filter ,that means the no. of stages used in the
design of analog filter.

8. Designing of Butterworth filter
Designing steps :
Let - Ap=Attenuation in pass band
As=Attenuation in stop band
Ωp=pass band edge frequency
Ωc=Cut off frequency
Ωs=stop band edge frequency
Step 1 :
 Calculation for frequency of analog filter.
a) For impulse invariance method ,
Ω - frequency of analog filter
ῳ - frequency of digital filter
T – sampling time
b) For bilinear transformation method ,

9. Step 2:
 Evaluate the order N –
Step 3 :
 Calculate cut-off frequency –
a) For impulse invariance method-