Downsampling

When considering a sampled sequence with zero values between sampling instants, one can replace it by taking every N-th value of the sequence. At these integer multiples of N, the original and sampled sequences coincide. This process, known as decimation, involves extracting every N-th sample from a sequence, thereby creating a more efficient sequence.

The Fourier transform of the decimated sequence reveals a combination of scaled and shifted versions of the original spectrum. This transformation focuses on the non-zero intervals of the sequence, simplifying analysis. The relationship between the Fourier transforms of the original and decimated sequences shows that the latter is a scaled version of the former, emphasizing the periodic nature introduced by decimation. The spectra of the decimated sequence differ from the original only in terms of frequency scaling.

If the original spectrum is band-limited and free of aliasing, decimation effectively spreads the spectrum over a larger frequency band. This spreading occurs because decimation reduces the sampling rate by a factor of N. To avoid aliasing, it is crucial that the original signal is oversampled, meaning the sampling frequency is sufficiently high relative to the signal's highest frequency component.

In practical terms, decimating a sequence derived from a continuous-time signal is also known as downsampling. This process reduces the data rate, making it more manageable while preserving essential characteristics of the original signal. When the original sequence is interpreted as samples from a continuous-time signal, careful consideration must be given to the sampling theorem to ensure no information loss due to aliasing.

Decimation is a valuable technique in digital signal processing, enabling more efficient data handling and analysis. By reducing the number of samples and maintaining critical spectral information, decimation allows for effective processing and transmission of signals in various applications, including telecommunications, audio processing, and data compression. Ensuring that the original signal is adequately oversampled before decimation is key to preventing aliasing and preserving the integrity of the reconstructed signal.