Designing FIR Filters for Signal Processing

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ASSIGNMENT 3
INTRODUCTION
Signals are obtained from the physical systems based on voltage values over a given time frame.
Disturbances and noise may affect the signal obtained. These disturbances are experienced from
the effects of the surroundings as well as the processing units [1]-[5]. Filters are used to attenuate
a selected range of frequencies from the continuous time signal. The design of filters is quite
crucial especially when the analog signal is under analysis so as to preserve as much of the
original signal as is possible. There are different types of filters within the transfer function with
a bandwidth of,
H ( f ) =|H ( f )|e−iθ ( f )
A designer is able to choose between the digital filter and the analog filter. A digital filter
has a very high accuracy, linear phase, no drift, easy to simulate and design, computation must
be completed within sampling period, and there is need for high speed ADC and DAC. On the
other hand, the analog filter is considered to be a bit less accurate as a result of the nonlinear
phase and the drift is as a result of the components [6]. The filter is difficult to simulate and
design as well as the limited by the op-amp or transistor gain bandwidth. Its design does not
require digital ICs. To determine the impulse response of the ideal low pass filter given the
signal is in frequency domain.

h ( t ) =I { H ( f ) }
¿ ∫
−∞
∞
H ( f ) e j 2 πft df
¿ ∫
− fu
fu
e− j 2 πf t0
e j 2 πft df
¿ ∫
−∞
∞
e j 2 πf (t −t0) df
¿ 2 f u
sin 2 π f u ( t−t0 )
2 π f u ( t−t0 )
Further, when performing impulse response on a signal, the FIR and IIR filters are used. The
finite impulse response, FIR, can be computed faster using the linear phase. There is no feedback
loop involved in the filter as it only has a feed-forward only architecture. Some of the items to
consider for the design are the windowing, the equirriple design is required as well as the
frequency selection. The filter uses the weighted least squares as well as the parks-Mcclellan
method to design the filter [7]. On the other hand, the infinite impulse response presents a slower
response when connecting to the feedback path, with a non-linear phase as well as it has a
feedback loop that is used to control the filter characteristics. The IIR filters focus on the bilinear
transformation and the impulse invariant are used to develop the design of an IIR filter. The FIR
filter is of the form,
y ( n ) = ∑
i=0
M −1
h ( i ) x ( n−i )
The transfer function is given as,

H ( z )= Y ( z )
X ( z ) =
∑
k =0
M
bk z−k
1+∑
k=1
N
ak z−k
For the fir filter, there is no feedback. The length is given as order M,
y ( n )= ∑
k=0
M −1
bk x ( n−k )= ∑
k=0
M −1
h ( k ) x ( n−k )
To determine the exact linear phase response of the fir filter, to determine the impulse response
of the ideal low pass filter,
h ( t ) =I { H ( f ) }
¿ ∫
−∞
∞
H ( f ) e j 2 πft df
¿ ∫
− fu
fu
e− j 2 πf t0
e j 2 πft df
¿ ∫
−∞
∞
e j 2 πf (t −t0) df
¿ 2 f u
sin 2 π f u ( t−t0 )
2 π f u ( t−t0 )
The low pass filter allows the low frequency components of a signal given below for the cutoff
frequency and denies all other frequency components above the cut off frequency. The filter
satisfies
H LP ( e jw )= { 10 ≤ ω ≤ ωc
0 , ωc ≤ ω ≤ π
The impulse response of the ideal low pass filter is given as,
hLP [n ]= sin ( ωc n )
πn
Windowing