What is filter ?
A filter is a circuit that passes certain frequencies and rejects all others.
What do we mean by passband ?
The passband is the range of frequencies allowed through the filter.
What is critical frequency fc ?
The critical frequency defines the end (or ends) of the passband. is normally specified at
the point where the response drops - 3 dB (70.7%) from the passband response.
What is a transition region ?
It is a region Following the passband leading into stopband region
What is a stop region ?
It is a region at the end of passband
Basic filter response
low-pass filter
 Basic low-pass filter Circuit
The low-pass filter allows frequencies below the critical frequency to pass and rejects
other. The simplest low-pass filter is a passive RC circuit with the output taken across C
 Ideal and Practical low-pass filter response
 Practical low-pass filter response
BW is the bandwidth of an ideal low-pass
Where : BW=fc
Example:
A certain low-pass filter has a critical frequency of 800 Hz. What is its bandwidth?
Calculation
Simple LP Digital filter
vin(t)xi
v
out(t)yi
dtt
dvoutyi-yi-1
That is, this discrete-time implementation of a simple RC low-pass filter is the exponentiallyweighted moving average
// Return RC low-pass filter output samples, given input samples, // time
interval dt, and time constant RC
function lowpass(real[0..n] x, real dt, real RC)
var real[0..n] y
var real α := dt / (RC + dt)
y[0] := x[0]
for i from 1 to n y[i] := α * x[i] + (1-α) * y[i-1]
return y
for i from 1 to n y[i] := y[i-1] + α * (x[i] - y[i-1])
High-Pass filter
Basic high-pass circuit
A high-pass filter is one that significantly attenuates or rejects all frequencies below fc and
passes all frequencies above fc. The critical frequency is, again, the frequency at which the
output is 70.7% of the input (or - 3 dB)
 Practical low-pass filter response
 Ideal and Practical low-pass filter response
Calculation
Simple HP Digital filter
// Return RC high-pass filter output samples, given input samples,
// time interval dt, and time constant RC
function highpass(real[0..n] x, real dt, real RC)
var real[0..n] y
var real α := RC / (RC + dt)
y[0] := x[0]
for i from 1 to n
y[i] := α * y[i-1] + α * (x[i] - x[i-1])
return y
for i from 1 to n y[i] := α * (y[i-1] + x[i] - x[i-1])
FIR (finite impulse response) Filters
• These filters are systems for which each output sample is the sum of a finite number
of weighted samples of the input sequence.
• a filter is a system that is designed to remove some component or modify some
characteristic of a signal, but often the two terms are used interchangeably.
• the basic input-output structure of the FIR filter as a time domain computation
based on a feed-forward difference equation.
Discrete-Time Systems
• A discrete-time system is a computational process for transforming one sequence,
called the input signal, into another sequence called the output signal
• In general, we represent the operation of a system by the notation
• Block-diagram representation of a discrete time system.
• Simple Example
In this case, the output depends only on one input value.
• complicated example would be the following system definition:
In this case, the output depends on three consecutive input values.
The Running-Average (moving average) Filter
• A simple but useful transformation of a discrete-time signal is to compute a moving
average or running average of two or more consecutive numbers of the sequence,
thereby forming a new sequence of the average values.
• Averaging is commonly used whenever data fluctuate and must be smoothed
prior to interpretation.
• Case study : stock-market prices fluctuate noticeably from day to day, or hour to
hour.
example
Finite-length input signal, x[n].
• The sequence is an example of a finite-length signal.
• we can compute a new sequence called y [n], which is the output of the
averaging operator.
• A 3-point average of the values {x[0], x[l], x[2]} = [2, 4, 6} gives the answer
1/3 (2 + 4 + 6) =4.
• The next output value is obtained by averaging (x[l], x[2], x[3]} = (4, 6, 4} which,
yields a value of 14/3.
• With this indexing, the equations for computing the output from the input are
• which generalizes to the following input-output equation:
• The equation given in is called a difference equation. It
Finite-length input signal, x[n].
Output of running-average filter, y[n]
to compute the entire output signal for all index values — < n <  . For the input of
x[n], the result is the signal y[n] tabulated as follows:
• Note that the values in orange type in the x[n] row are the numbers involved in the
computation of y[2].
• Also note that y[n] = outside of the finite interval —2 < n < 4; i.e., the output also has
finite support.
Observe that the output sequence is longer (has more nonzero values)
than the input sequence, and that the output appears to be a somewhat rounded-off
version of the input; i.e. it is smoother than the input sequence.
In this case, n would stand for time, and we can interpret y[n] . as the computation of the
present value of the output based on three input values. Since these inputs are indexed as
n, n + 1, and n + 2, two of them are "in the future."
Causal And Noncasual Filter
• In general, values from either the past or the future or both may be used in the
computation.
• The running-average filter calculation at the present time (l=n) uses values within a
sliding window.
• Gray shading indicates the past (L<n); orange shading, the future (L > n). Here, the
sliding window encompasses values from both the future and the past.
• as shown in the figure . In all cases of a 3-point running average, a
• sliding window of three samples determines which three samples are used in
the computation of y[n].
• A filter that uses only the present and past values of the input is called a causal filter.
backward average
• a filter that uses future values of the input is called noncausal.
• Noncausal systems cannot be implemented in a real-time application because the input
is not yet available when the output has to be computed. In other cases, where stored
data blocks are manipulated inside a computer, the issue of causality is not crucial.
• Observe that the output of the causal filter is simply a shifted version of the output
of the previous noncausal filter.
EXERCISE 5.1: Determine the output of a centralized average for the following figure
Is this filter causal or noncausal?
What is the support of the output for this input?
How would the plot of the output compare to the figure below ?
FIR filter in Labview
The General FIR Filter
In the previous example ( not the exercise ): M = 2 and bk = 1/3 for k = 0, 1, 2,
• If the coefficients bk are not all the same, then we might say that y[n] defines a weighted
running average of M + 1 samples.
Operation of the Mth order causal FIR filter showing various positions of the sliding
window of M + 1 points under which the weighted average is calculated. When the input
signal x [n] is also finite length (N points), the sliding window will run onto and off of the
input data, so the resulting output signal will also have finite length
The parameter M is the order of the FIR filter
The number of filter coefficients is also called the filter length (L).
L =M+1
An Illustration of FIR Filtering
consider a signal
Unit Impulse Sequence
The unit impulse is perhaps the simplest sequence because it has only one nonzero value,
which occurs at n = 0. The mathematical notation is that of the
Kronecker delta function
It is tabulated in the second row of this table:
Shifted impulse sequence
• A shifted impulse such as [n - 2] is nonzero when its argument is zero, i.e., n - 2 = 0,
or equivalently n = 2. The third row of the table gives the values of the shifted impulse
[n-2].
• The shifted impulse is a concept that is very useful in representing signals and systems.
Consider, for example, the signal
The following table shows the individual sequences and their sum:
Block diagram showing definition of impulse response
any sequence can be represented in this way.
• The equation is true if k ranges over all the nonzero values of the sequence x[n].
• The equation states the obvious: The sequence is formed by using scaled shifted
impulses to place samples of the right size at the right positions.
Unit Impulse Response Sequence
When the input to the FIR filter is a unit impulse sequence, x[n] = [n], the output is, by
definition, the unit impulse response, which we will denote by
h[n]. Substituting x[n] = [ n] in
gives the output y[n] = h[n]:
• the impulse response h[n] of the FIR filter is simply the sequence of difference equation
coefficients.
EXERCISE 5.3: Determine and plot the impulse response of the FIR system
The Unit-Delay System
One important system is the operator that performs a delay or shift by an amount n0
Delayed Finite-length input signal , y[n]=x[n-2]
• When n0 = 1, the system is called a unit delay.
• The delay system is actually the simplest of FIR filters; it has only one nonzero
coefficient.
For example, a system that produces a delay of 2 has filter coefficients
{bk } = [0, 0, 1}. The order of this FIR filter is M = 2, and its difference equation is