FIR digital filters
FIR Digital Filter
One of the most important properties of FIR digital filters is that they can realize a frequency response with a linear phase. Recognizing that a linear phase response corresponds to a constant delay, and can greatly simplify the approximation problem in the design of FIR digital filters. Specifically, the design simplifies to that of approximating the desired magnitude response.
The main lobe of a window w[n] is defined as the frequency band between the first zero crossings of its magnitude response |W(e^jΩ| on either side of the origin. The parts of the magnitude response that lie on either side of the main lobe are referred to as side lobes. The width of the main lobe and the amplitude of the side lobes provide measures of the extent to which the frequency response W(e^jΩ) deviates from an impulse function located Ω=0.
The oscillations, a consequence of the sidelobes in |W(e^jΩ|, may be reduced by using a different window with smaller sidelobes. A practical window commonly used for this purpose is known as the Hamming window.
Some important features to keep in mind about the Hamming window are as follows:
- The main lobe of the rectangular window is less than half the width of the main lobe of the Hamming window.
- The side lobe of the Hamming window, relative to the main lobe, is greatly reduced compared with those of the rectangular window. Specifically, the peak amplitude of the first side lobe of the rectangular window is only about 13 dB below that of the main lobe, whereas the corresponding value for the Hamming window is 40dB.
This happens as the Hamming window reduces oscillations in the frequency response of an FIR digital filter.
Filtering of speech signals
The preprocessing of speech signals is fundamental to many applications, such as the digital transmission and storage of speech, automatic speech recognition, and automatic speaker recognition systems. FIR digital filters are used for such systems.
- It is essential to maintain precise time alignment in speech processing applications. A finite-duration impulse response or FIR digital filter helps fulfill this requirement.
- The approximation problem in filter design is greatly simplified by the exact linear phase property of an FIR digital filter. In particular, in not having to deal with delay distortion, our only concern is that of approximating the desired magnitude response.
IIR Digital Filters
One of the common methods used for the conversion of analog transfer functions to digital transfer functions is based on bilinear transform and is widely used in the IIR digital filter’s design.
The properties of bilinear transform are as follows:
- In the bilinear transform, the left half of the s-plane is mapped into the interior of the unit circle in the z-plane.
- The whole jw-axis of the s-plane is inscribed into one complete revolution of the unit circle in the z-plane.
- Also, the right half of the s-plane is mapped into the exterior of the unit circle in the z-plane.
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