IntroductionFilters that pass certain spectral components of the input signal without distortion and completely attenuate the remainder of the spectral components are known as frequency selective or signal separation filters. Whereas filters that change the shape of the spectrum are called frequency shaping or signal restoration filters. To learn more about filters click here. Digital Filters operates on digital signals (sampled version of the signals) and basically computes a sequence of numbers (input signal) to produce another sequence of numbers (filtered output signals). It is governed by linear difference equation. To learn more about digital filters click here. Digital Filters are broadly classified as FIR Filters and IIR Filters.
Filters designed by using finite number of samples of impulse response are called FIR filters. An array of the most recent data samples is multiplied by an array of constants (called the tap coefficients), and then summed (results in dot product). The process is repeated for next sample of data. To learn more about FIR Filters click here. Design techniques for linear phase FIR filters are
1. Fourier Series Method and Window method
Frequency response of a digital filter is periodic (periodicity equal to sampling frequency) and can be expressed as linear combination of complex exponentials for a finite duration (since FIR filters have finite number of samples of impulse response). This abrupt truncation of fourier series to finite samples results in oscillations in passband and stopband, known as Gibbs Phenomenon. This undesirable oscillations is reduced my multiplying desired impulse response coeffecients by a window function. To learn more about Gibbs phenomenon and fourier series methodology of FIR filter design click here. Frequency Sampling methodDesired frequency response is known and the given frequency response is sampled at a set of equally spaced frequencies to obtain N samples basically resulting in N-point DFT. Using the IDFT formula, the filter co-efficients is calculated. The continuous frequency response is calculated as an interpolation of the sampled frequency response.The major disadvantage of frequency sampling method is that frequency response obtained by interpolation is equal to the desired frequency response only at the sampled points resulting in zero approximation error at those points and some finite error at other points. However, this finite error could be reduced by increasing the number of samples. To learn more about FIR Design by Frequency Sampling method click here. To read more examples about FIR design using frequency sampling method and Discrete Fourier Transform click here.
For MATLAB code for FIR Filter design and other DSP operations go to
Filters that has an infinite duration of impulse response are called Infinite Impulse Response filter or IIR filter. An Infinite Impulse Response (IIR) filter produces an output that is the weighted sum of the current and past inputs and past outputs.To learn more about IIR Filters click here. To learn about the basic design methodology of various IIR filter responses click here. Depending on the amplitude, group delay and ripple characteristics, IIR filter responses is further classified into four types:
1. Butterworth (flat amplitude)
Butterworth filter responses can be achieved using passive components like resistors, inductors and capacitors and is governed by the transfer function involving a parameter n, called the order of the filter. It is also called the Maximally Flat approximation, because for a response of order n, the first (2n-1) derivatives of the gain with respect to frequency are zero at frequency = 0. There is no ripple in the passband, and DC gain is maximally flat.
To learn about the transfer function, poles and characteristics of Butterworth filters click here.
To learn more about Butterworth Filters - design aspects and examples using Matlab click here.
Bessel filter responses are characterised by flat or constant group delay in the pass band. The major drawback is that the Bessel filter has the slowest rolloff and requires the highest order to meet an attenuation specification. To learn about Bessel analog filter design functions using Matlab click here. To review the governing equations for phase transfer, amplitude, phase and magnitude response and group delay for various order of Bessel filter click here.
Chebychev filter response has equiripple in passband or stopband.It is also refered to as an equal-ripple approximation.Superior attentuation in the stop band is achieved at the expense of ripple in the passband. There are two types of Chebychev filter responses, viz type I and type II. Type I have ripple in pass band and Type II have ripple in stop band.To learn more about Chebychev filter response click here.
Elliptic filter responses are based on elliptic functions.Elliptic filters have the most rapid transition (narrow transition band) but they ripple in both pass band and stop band. To learn more about Elliptical filter response click here.
For class notes on Chebyshev and Butterworth IIR Filters click here.
last updated: August 24, 2003 |