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Fundamentals of Digital Signal Processing

Spectrum After the Time Shift II

Explore the impact of time shifts on digital signals in the frequency domain using Fourier transform principles. Understand how shifting a unit impulse affects its discrete Fourier transform spectrum, producing frequency-domain complex sinusoids. This lesson guides you through the mathematical derivation and application of these concepts to model and analyze signals effectively.

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In this chapter, we’ll look at the time shift from an IQIQ viewpoint.

A shift in impulse

Let’s assume that a unit impulse signal δ[n]\delta[n] is time-shifted by n0n_0. From the DFT definition, we get

X[k]=∑n=0N−1δ[n−n0]e−j2πkNn=e−j2πkNn0X[k]=\sum_{n=0}^{N-1}\delta[n-n_0]e^{-j2\pi\frac{k}{N}n}=e^{-j2\pi\frac{k}{N}n_0}

because δ[n−n0]\delta[n-n_0] is 11 at ...