How to Shift a Signal in Time Domain

When learning DSP, it’s important to know how to shift a signal in time. This is an important part of all DSP operations.

• Some applications require signals to be shifted by a processor. For example, a signal needs to be aligned with the output of another signal processing chain.
• In some other applications, we encounter a signal shifted in time by nature. For example, a localization receiver attempts to estimate the time shift of the received signal to compute its range from the transmitter.
• In almost all DSP operations, like filtering, a signal needs to be shifted in time to compute the output response of the system.

The logic

For a discrete-time signal $x[n]$, its delayed version by $m$ units is denoted by $x[n-m]$. Taking the time $n$ as the present moment, we need to go $m$ samples in the past on the time axis.

On the other hand, its advanced version by $m$ units is denoted by $x[n+m]$. Taking the time $n$ as the present moment, we need to go $m$ samples into the future on the time axis.

In both cases, our focus should be on the time axis. An example of a delay and advance by 1 unit of time is plotted in the figure below. It’s clear that the time axis is pulled back by 1 unit in the second subfigure while it’s advanced by 1 unit in the third subfigure. If this sounds confusing, check the time index $n=0$ in all subfigures.