Introduction
We start by considering the issue of distortionless transmission, which is basic to the study of linear filters and equalizers. This will lead to a discussion of an idealized framework for filtering, which, in turn, provides the basis for the design of practical filters. The design of a filter can be accomplished by using continuous-time concepts, in which case we speak of analog filters. Alternatively, the design can be accomplished by using discrete-time concepts, in which case we speak of digital filters. Analog and digital filters have their own advantages and disadvantages.
Consider a continuous-time LTI system with impulse response h(t). Equivalently, the system may be described in terms of its frequency response H(jw), defined as the Fourier transform of h(t). Let a signal x(t) with Fourier transform X(jw) be applied to the input of the system. Let the signal y(t) with the Fourier transform Y(jw) denote thge output of the system. To learn about the conditions for distortionless transmission through the system. By “distortionless transmission” we mean that the output signal of the system is an exact replica of the input signal, except, possibly for two minor modifications.
- A scaling amplitude
- A constant time delay.
Ideal Low- Pass Filters
Typically, the spectral content of an information-bearing signal occupies a frequency band of some finite extent. For example, the spectral continents of a speech signal essential for telephonic communication lie in the frequency band from 300 to 3100 Hz. To extract the essential information content of a speech signal for such an application, we need a frequency selective system filter that limits the spectrum of the signal to the desired band of frequencies. Indeed, filters are basic to the study of signals and systems, in the sense that every system used to process signals contains a filter of some kind in its composition.
The frequency response of a filter is characterized by a passband and a stopband, which are separated by a transition band, also known as a guard band. Signals with frequencies inside the passband are transmitted with little or no distortion, whereas those with frequencies inside the stopband are effectively rejected. The filter may thus be of the low-pass, high-pass, band-pass, or band-stop type, depending on whether it transmits low, high, intermediate, or all but intermediate frequencies, respectively.
Consider, then, an ideal low-pass filter, which transmits all the low frequencies inside the passband without any distortion and rejects all the high frequencies inside the stopband. The transition from the passband to the stopband is assumed to occupy zero width. In so far as low-pass filtering is concerned, the primary interest is in the faithful transmission of an information-bearing signal whose spectral content is confined to some frequency band defined by 0 <= omega <= omega_{c}. Accordingly, in such an application, the conditions for distortionless transmission need to be satisfied only inside the passband of the filter.
The inverse relationship that exists between the two parameters:
(1) the duration of the rectangular input pulse applied to an ideal low-pass filter and
(2) the cutoff frequency of the filter.
This inverse relationship is a manifestation of the constancy of the time-bandwidth product. From a practical perspective, the inverse relationship between pulse duration and filter cutoff frequency has a simple interpretation, in the context of digital communications. If the requirement is merely that of recognizing that the response of a low-pass channel is due to the transmission of the symbol 1, represented by a rectangular pulse of duration T0 , it is adequate to set the cutoff frequency of the channel.
Design of filters
The low-pass filter with frequency response is “ideal” in that it passes all frequency components lying inside the passband with no distortion and rejects all frequency components lying inside the stopband, and the transition from the passband to the stopband is abrupt. Recall that these characteristics result in a non implementable filter. Therefore, from a practical perspective, the prudent approach is to tolerate an acceptable level of distortion by permitting prescribed “deviations” from these ideal conditions, as described here for the case of continuous-time or analog filters.
Analogous specifications are used for discrete-time filters, with the added provision that the response is always 2pi periodic in n. So long as these specifications meet the goal for the filtering problem at hand and the filter design is accomplished at a reasonable cost, the job is satisfactorily done.
Having formulated a set of specifications describing the desired properties of the frequency-selective filter, we set forth two distinct steps involved in the design of the filter, pursued in the following order:
The specifications just described favor an approach that focuses on the design of the filter based on its frequency response rather than its impulse response. This is in recognition of the fact that the application of a filter usually involves the separation of signals on the basis of their frequency content.
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