Signal Analysis & Synthesis

Analogue Modulation

Analogue Signal Processing

Digital signal processing

Digital modulation

Communication Systems

Real and Ideal Filters

Filter advanced network theory: It has been found that ideal filters cannot be physically realized. Consider, for example, the impulse response of an ideal LPF. By definition and using

This is a sin c pulse in time and non-zero for t < 0.

we observe: the output h(t) appears before the impulse is applied. Such a filter is said to be anticipatory, and the portion of the output appearing before the input is called a precursor. Clearly, such behavior is physically impossible, and hence the filter must be non-realizable. Similar results are found for the band-pass and high-pass case.

However, ideal filters are conceptually useful in the study of communication systems. In practice, filters can be designed which come quite close to being ideal, at least for engineering purposes. The above illustration shows a practical low-pass filter which is both simple and has a reasonably sharp cutoff. Like all real filters, the cut-off is not perfectly straight, so the bandwidth is internationally specified in terms of the 3 - dB frequency points. The impulse response shown in above illustration is seen to be similar to a line pulse minus the precursors.

Filters with more complicated designs, such as Butterworth and Chebyshev filters, more closely approximate the ideal filter.

We further observe:

• As the number of reactive elements increases without limit, the transfer function can be made arbitrarily close to that of an ideal filter.
• But at the same time, the filter time delay increases without limit, making the filter useless.
• Moreover, the infinite time delay means the precursors, will always appear after the input is applied, which must be true of a real filter.

In our further analysis, we shall often assume filters to be virtually ideal and examine their effects on signal transmission. We should keep in mind that in the frequency domain, the effects are obvious as frequency components falling outside the filter pass-band are removed from the spectrum but effects in the time domain may or may not be easily captured.