Downsampling

Consider a sampled sequence with zero values between sampling instants. Replace it by taking every N-th value of the sampled sequence.

The original and sampled sequences are equal at integer multiples of N.

Decimation extracts every N-th sample from a sequence, making the new sequence more efficient.

The Fourier transform of the decimated sequence is a combination of scaled and shifted versions of the original spectrum.

This transform simplifies analysis by focusing on non-zero intervals.

The final relationship shows the Fourier transform of the decimated sequence is a scaled version of the original's transform.

This scaling emphasizes the periodic nature introduced by decimation, with spectra differing only in frequency scaling.

If the original spectrum is band-limited with no aliasing, decimation spreads the spectrum over a larger frequency band.

Decimating a sequence from a continuous-time signal reduces the sampling rate by a factor of N, avoiding aliasing if the original signal is oversampled.

When interpreting the original sequence as samples from a continuous-time signal, decimation is called downsampling.