Sampling

Sampling signals for digital use turns a continuous-time signal, x(t), into a discrete-time signal. This step is essential when a real-world signal must be stored or processed by a digital system. A train of impulses sets the sampling interval, and the sampling frequency describes how often the signal is measured.

The sampling process works by multiplying the continuous-time signal by the impulse train. This creates a series of discrete impulses called the sampled signal. Engineers then study...

Impulse-train sampling and zero-order hold are two key steps in digital signal processing. Together, they show how a continuous-time signal can be turned into samples and then built back into a continuous-looking waveform.

Impulse-train sampling uses a periodic impulse train. This train is a series of delta functions placed at regular intervals set by the sampling period. When the continuous-time signal is multiplied by the impulse train, the result is a set of impulses whose amplitudes match...

Signal reconstruction with low-pass filtering helps turn a sampled signal back into a smooth continuous signal. When a continuous signal is sampled with a period T, the sampled version creates replicas of the original spectrum in the frequency domain. These replicas are spaced by the sampling frequency.

A zero-order hold can be used to process the sampled signal. This method keeps each sample value until the next sampling period, so the output becomes piecewise constant. In the frequency...

Sampling a signal at the right rate is essential for accurate signal reconstruction. A time-domain signal can be analyzed with its Fourier transform, which reveals its spectrum. When the signal is sampled, the spectrum becomes multiple scaled replicas in the frequency domain. The spacing between these replicas depends on the sampling frequency.

If the sampling frequency is below the Nyquist rate, the replicas overlap. The Nyquist rate is twice the highest frequency in the original signal. Once...

Decimation reduces a sampled sequence by keeping every N-th sample. At those integer multiples of N, the original and reduced sequences match. This makes the sequence shorter and more efficient for digital signal processing.

The Fourier transform of a decimated sequence is not just a simple copy of the original. It becomes a scaled and shifted combination of the original spectrum. In effect, the reduced sequence reflects the same frequency content, but in a form shaped by the new sampling...

Zero-inserted signal reconstruction is a digital signal processing method used to raise a signal’s sampling rate while keeping its important frequency content. A decimated signal has fewer samples and a narrower frequency range. To upsample it, zeros are inserted between the original samples.

This zero-stuffed sequence changes the spectrum. It creates repeated spectral replicas at intervals set by the new Nyquist frequency, which is the highest frequency allowed by the new sampling rate. A...

Bandpass sampling is a signal processing method for signals whose energy is concentrated in a narrow frequency band. Such a signal is called a bandpass signal. The method works when the sampling rate is greater than twice the signal bandwidth, which helps prevent aliasing.

A bandpass signal has a spectrum with a lower limit, ω1, and an upper limit, ω2. Its spectrum is nonzero only between those two frequencies. To sample it, the time-domain signal is multiplied by an impulse train. This...