Mr.Nilesh Rambahadur Jaiswar
Mr.Kunal Dilip Vartak
In signal processing, the function of a filter is to
remove unwanted parts of the signal, such as
random noise, or to extract useful parts of the
signal, such as the components lying within a
certain frequency range.
Operation Of Digital Filters :
• Raw Signal : V = x(t) Where t is time.
• This signal is sampled at time intervals h (the sampling interval).
The sampled value at time t = ih is xi=x(ih)
• Transferred From The ADC To The Processor
x0 , x1 , x2 , x3 , ...
• Corresponding to the values of the signal waveform at
t = 0, h, 2h, 3h, ...
• And t = 0 is the instant at which sampling begins.
• At time t = nh the values available to the processor
x0 , x1 , x2 , x3 , ... Xn
• The digital output from the processor to the DAC
y0 , y1 , y2 , y3 , ... yn
Examples Of Simple Digital Filters
Unity Gain Filter:
yn = xn
Each output value yn is exactly the same as the corresponding
input value xn: y0 = x0 , y1 = x1 , .................
Simple Gain Filter:
yn = Kxn
where K = constant.
This simply applies a gain factor K to each input value.
Pure Delay Filter:
yn = xn-1
The output value at time t = nh is simply the input at time
t = (n-1)h , i.e. the signal is delayed by time h : y0 = x-1 ,y1 = x0,.....
Two-term Difference Filter:
yn = xn– xn-1
The output value at t = nh is equal to the difference between
the current input xn and the previous input xn-1 :
y0 = x0– x-1 ,
y1 = x1– x0 ,
y2 = x2– x1 , ... etc
Two-term Average Filter:
yn = ( xn + xn-1 )/2
The output is the average (arithmetic mean) of the current
and previous input:
y0 =(x0 + x-1)/2 ,
y1 =(x1 + x0)/2 ,
y2 =(x2 + x1)/2 , ... etc
Three-term Average Filter:
yn =(xn + xn-1 + xn-2)/ 3
This is similar to the previous example, with the average being
taken of the current and two previous inputs:
y0 =(x0 + x-1 + x-2)/ 3
y1 =(x1 + x0 + x-1)/ 3
y2 =(x2 + x1 + x0)/ 3
As before, x-1 and x-2 are taken to be zero.
Central Difference Filter:
yn =(xn– xn-2)/ 2
The output is equal to half the change in the input signal over
the previous two sampling intervals:
y0 =(x0 – x-2)/ 2
y1 =(x1 – x-1)/ 2 , etc….
Types of Digital Filters
Finite Impulse Response , Or FIR Filters :
This filters express each output sample as a weighted
sum of the last N input samples, where N is the order
of the filter. FIR filters are normally non-recursive,
meaning they do not use feedback and as such are
Infinite Impulse Response , Or IIR Filters:
This filters are the digital counter part to analog
filters. Such a filter contains internal state, and the
output and the next internal state are determined by a
linear combination of the previous inputs and outputs
Finite impulse response
In signal processing , a finite impulse response (FIR)
filter is a filter whose impulse response (or response to any
finite length input) is of finite duration, because it settles
to zero in finite time.
Are Inherently Stable
This is due to the fact that, because there is no required
feedback, all the poles are located at the origin and thus are
located within the unit circle .
Require No Feedback:
This means that any rounding errors are not compounded
by summed iterations.
The same relative error occurs in each calculation.
This also makes implementation simpler .
They Can Easily Be Designed to be linear phase by making
the coefficient sequence symmetric; linear phase, or phase
change proportional to frequency, corresponds to equal delay at
The impulse response h[n]can be calculated if we set x[n]= δ[n]in
the above relation, where δ[n] is the Kronecker delta impulse. The
impulse response for an FIR filter then becomes the set of
coefficients , as follows
For n=0 to N
The Z-transform of the impulse response
FIR filters are clearly bounded - input
bounded-output (BIBO) stable, since
the output is a sum of a finite number of
finite multiples of the input values, so can
be no greater than
To design a filter means to select the coefficients
such that the system has specific characteristics.
Window design method
In the Window Design Method, one designs an ideal IIR filter,
then applies a window function to it –Ve in the time domain,
multiplying the infinite impulse by the window function. This
results in the frequency response of the IIR being convolved
with the frequency response of the window function. If the
ideal response is sufficiently simple, such as rectangular, the
result of the convolution can be relatively easy to determine.
Frequency Sampling method
Weighted least squares design
Moving Average Example
Block diagram of a simple FIR filter
(2nd-order/3-tap filter in this case,
implementing a moving average)
A moving average filter is a very simple FIR filter. It is
sometimes called a boxcar filter.
The filter coefficients, b0,......,bN are found via the
To provide a more specific example, we select the filter
The impulse response of the resulting filter is:
z-transform of the impulse response:
Infinite Impulse Response
Infinite Impulse Response (IIR) is a property of signal
processing systems. Systems with this property are known as IIR
systems or, when dealing with filter systems, as IIR filters.
Implementation And Design
IIR filters may be implemented as either analog or digital
filters. In digital IIR filters, the output feedback is
immediately apparent in the equations defining the
output. Note that unlike FIR filters, in designing IIR
filters it is necessary to carefully consider the "time zero"
case in which the outputs of the filter have not yet been
Transfer Function Derivation
P is the feedforward filter order
bi are the feedforward filter coefficients
Q is the feedback filter order
ai are the feedback filter coefficients
x[n] is the input signal
y[n] is the output signal.
IIR Filter Z Transfer Function
Description Of Simple IIR Filter Block Diagram
The transfer function allows us to judge whether or not a
system is bounded-input, bounded-output (BIBO) stable. To
be specific, the BIBO stability criteria requires that the
ROC(Radius Of Convergence) of the system includes the unit
circle. For example, for a causal system, all poles of the
transfer function have to have an absolute value smaller than
one. In other words, all poles must be located within a unit
circle in the Z-plane.
The poles are defined as the values Z of which make the
denominator of H(z) equal to 0 :
If aj =0 then the poles are not located at the origin of the
Applications of Digital filter
Audio Systems Such As CD / Dvdplayers
Image Processing And Enhancement
Processing Of Seismic And Other Geophysical
Processing Of Biological Signals
It has linear phase response.
Thermal and environmental variation cannot change the
It is possible to filter several input sequences without any
Actually the speed operation totally depends on the
number of the arithmetic operation in the processor.
Finite word-length effect, which results quantizing noise
and round-off noise.
It needs much longer time to design and develop the digital
sequences though it can be used on other tasks or
applications once developed.
The main utility of the analysis methods presented is in
ascertaining how a given filter will affect the spectrum of a
signal passing through it. Some of the concepts introduced
were linearity, time-invariance, filter impulse response,
difference equations, transient response, steady-state
response, transfer functions, amplitude response, phase
response, phase delay, group delay, linear phase, minimum
phase, maximum phase, poles and zeros, filter stability, and
the general use of complex numbers to represent signals,
spectra, and filters.
Under Guidance of
Ms. Pradnya Vartak
Mr. Nilesh Rambahadur Jaiswar
Mr.Kunal Dilip Vartak