# Linear Distortion and Signal Bandwidth

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Strictly speaking, linear distortion applies only to linear time-invariant (LTI) systems. An LTI system with linear distortion is called dispersive. If the system bandwidth is much greater than the signal bandwidth, then the system is approximately distortion free. In this Demonstration, the signal is a casual unit pulse, and the corresponding signal bandwidth is 1 Hz. The system is a lowpass RC filter, and the 3 dB bandwidth is (Hz). By displaying the pulse response and the associated signal spectrum and system transfer function, this Demonstration shows how the system approaches an ideal system as increases.

Contributed by: Victor S. Frost (August 25)

(University of Kansas)

Open content licensed under CC BY-NC-SA

## Details

An LTI system is distortion free (ideal) if for an input signal the output is given by with . The corresponding ideal transfer function is ; that is, a constant amplitude and linear phase response. An LTI system is approximately ideal if the bandwidth of the system is much greater than the bandwidth of the signal. In this Demonstration, the input , a causal unit step with , and the first zero of is at Hz, so here the signal bandwidth is 1 Hz. The system is a lowpass RC filter with ; the 3 dB bandwidth of (Hz):

.

This Demonstration compares to in one plot and to in the other plot as you vary the ratio . For a large ratio, the Demonstration shows that .

Reference

[1] F. T. Ulaby and A. E. Yagel, *Signals and Systems: Theory and Applications*, Michigan Publishing, 2018. (Jul 21, 2023) ss2.eecs.umich.edu.

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