Part lV-Synthesis of FIR filters_compressed.pdf

DSP

1. Synthesis of FIR filters HETT208
Conception of digital filters
 given the transfer function H(z) and the impulse response h(n), the difference equation can
be obtained
 this difference equation could be implemented by a computer program or a specified
program integrated circuit
 synthesis of the digital filters begins with the definition of the filter characteristics
 to synthesize the digital filter, a known analog filter template is used to find a digital system
characterized by the transfer function H(z)
 determination of the transfer function by a direct method is not simple, on the other hand
transforming an analog filter to a digital filter is relatively simple
 many methods are based on designing a digital filter from the equivalent analog filter
 the objective of filter design is to find a stable function that is realizable using a suitable
filter structure top estimate a specified frequency response or impulse response
FIR and IIR
 two classes of filters are defined based on the length of their linear impulse response
 FIR, finite impulse response filters are always stable and can be designed to have exactly
linear phase
 IIR, infinite impulse response filters are unstable and can’t be designed to have linear phase
Synthesis of FIR filters
 FIR filters are attractive with the following advantages:
i. Unconditional stability(all poles en zero)
ii. Possible linear phase
 however it has some disadvantages:
i. requires greater number of coefficients than IIR filters to obtain the same frequency
features because of the absence of poles out of zero
 any stable and causal digital filter function can be approached by the transfer function of an
FIR filter
 output of an FIR filter can be expressed as a linear combination of a finite set of input
elements
 the output depends only on the inputs

2. Synthesis of FIR filters HETT208






 the weighing coefficients are therefore nothing but the values of the impulse response of
the filter
 two most used methods for the approximation of FIR filters are:
i. The window method
ii. The frequency sampling method
The window method
 technique consists of knowing the frequency response H(f) of the continuous frequency
response to be approached
 the impulse response is to be determined using the inverse of the Fourier transform
















 an ideal filter has an infinite length impulse response therefore non causal and unrealizable
because the filter will be unstable
 for an ideal filter is of range
 the design process involves shortening the filter to a desired length (truncation or
windowing), then delaying it to make it causal
 truncation is achieved by multiplication by the window function
 the truncated impulse response
has range
 when truncated thus
, the ideal filter originally rectangular shows
oscillatory behavior known as the Gibbs Phenomenon
 to make the filter causal we delay the impulse response by N thus

 delaying the samples doesn’t change the magnitude response but changes the phase
response

3. Synthesis of FIR filters HETT208
 using the modulation property of the DTFT, we find
which is the convolution in the
frequency domain



Ideal impulse responses of filters


Window functions

The frequency sampling method
 this method is relatively simpler and makes it possible to carry out any form of filter which
can’t be done by the window method

4. Synthesis of FIR filters HETT208
 this method is applied when the frequency response H(f) of an ideal continuous filter is
unknown
 the impulse response therefore can’t be calculated by the inverse Fourier transform and
the inverse DFT is used instead
 the desired response in the frequency domain is sampled, N points of this frequency
response obtained are made equivalent to N points of the temporal response to be
obtained by the inverse DFT
 we start by sampling ,












 we then apply the inverse DFT