What is the effect of the delay operator Z^{-1} on a discrete-time signal x[n]?

What is the effect of the delay operator Z^{-1} on a discrete-time signal x[n]?

• A. Advances the signal by one sample.
• B. Multiplies the signal by z.
• C. Delays the signal by one sample in time.
• D. Shifts the frequency spectrum.

Answer: (C)

The delay operator Z^{-1} shifts the signal forward in time by one sample. If you apply it to x[n], the result is y[n] = x[n−1], meaning each output sample is the previous input sample. In the Z-domain this is Y(z) = z^{-1} X(z), which is exactly the unit delay relationship. So the primary effect is a one-sample time delay. In frequency terms, delaying by one sample introduces a phase shift e^{-jω}, not a change in magnitude. The other options don’t match this behavior: advancing by one sample would give y[n] = x[n+1], and multiplying by z describes a different, transform-domain operation.

The delay operator Z^{-1} shifts the signal forward in time by one sample. If you apply it to x[n], the result is y[n] = x[n−1], meaning each output sample is the previous input sample. In the Z-domain this is Y(z) = z^{-1} X(z), which is exactly the unit delay relationship. So the primary effect is a one-sample time delay. In frequency terms, delaying by one sample introduces a phase shift e^{-jω}, not a change in magnitude. The other options don’t match this behavior: advancing by one sample would give y[n] = x[n+1], and multiplying by z describes a different, transform-domain operation.