On the Maximum Frequency for Discrete-Time Signals

For anyone working with discrete-time signals, it is almost second nature to know that the highest frequency is always equal to fs/2 Hz, where fs is the sampling rate. This is typically demonstrated by analyzing the sampling process in the frequency domain, where the signal spectrum gets replicated every fs Hz, causing all frequencies outside the range of [-fs/2 fs/2] to "fold back" into the main Nyquist range of [−fs/2 fs/2] Hz.

However, based on my experience with students in signal processing or DSP classes—and even with some DSP practitioners—many people don’t develop strong time-domain intuition for why the frequency in discrete-time signals can never exceed fs/2 Hz.

From continuous-time signal processing, students generally understand that faster oscillations in the time domain correspond to higher frequencies, i.e., as the signal oscillates faster and faster in time, its frequency increases higher and higher, and can go all the way to infinity. So what happens with the discrete-time signals that limit their max frequency to fs/2 Hz?

To build this intuition in the time-domain for discrete-time signals, I typically start by showing the example x[n]=cos(πn)=(−1)^n, and I ask them if they can imagine any other discrete-time signal that can oscillate any faster in time.

It immediately becomes obvious that the answer is no! And since this signal has a period of 2 samples, its frequency is 1/2 cycles/sample, which is indeed the maximum possible frequency. Now, if we simply introduce the sampling time Ts as the actual time in seconds between two consecutive samples, this signal would have a period of 2Ts seconds and hence the frequency of 1/2Ts=fs/2 Hz.  

This simple example highlights the fact that the speed of time-domain oscillations in discrete-time signals is limited by the fact that the samples are disjoint in time, and the signal is undefined between each two consecutive samples. And for rate of oscillations and thus the maximum frequency to increase, the samples must get closer to each other, i.e., the sampling rate fs=1/Ts should increase.

This helps solidify the understanding that discrete-time signals cannot oscillate faster than fs/2, bridging the gap between time-domain intuition and frequency-domain analysis.

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