18.7:
Bandpass Sampling
Consider a signal sampled at a rate greater than twice its highest frequency.
A signal with most of its energy concentrated in a narrow frequency band is known as a bandpass signal. The technique used for sampling such signals is called bandpass sampling.
The signal's spectrum has a lower frequency limit, ω1, that is greater than its bandwidth.
The spectrum is nonzero within the defined lower and upper frequency bands.
Multiplying the time-domain signal by an impulse train produces a sampled signal.
The spectrum of the sampled signal is obtained by convolving the signal's spectrum with the impulse train spectrum over 2π.
This convolution causes the signal's spectrum to repeat periodically with a period of 2π/T.
As T increases, the spacing between the repetitions decreases. Aliasing occurs when these repeated spectra overlap.
Aliasing can be avoided by ensuring the spacing between the repeated spectra is greater than the signal's bandwidth and with the maximum value of T.
Filters with specific constants and frequencies pass the frequencies within the desired range.
18.7:
Bandpass Sampling
In signal processing, bandpass sampling is an effective technique for sampling signals that have most of their energy concentrated within a narrow frequency band. This type of signal is known as a bandpass signal. The key principle of bandpass sampling involves sampling the signal at a rate that is greater than twice the signal's bandwidth to prevent aliasing.
A bandpass signal has a spectrum with a lower frequency limit, denoted as ω1, and an upper frequency limit, denoted as ω2. The spectrum is nonzero only within this specific frequency range. To sample this signal, the time-domain signal is multiplied by an impulse train, resulting in a sampled signal. The spectrum of this sampled signal is then obtained by convolving the signal's original spectrum with the spectrum of the impulse train over a range of 2π. This convolution process causes the signal’s spectrum to repeat periodically, with a period of 2π/T. Here, T is the sampling interval, and as T increases, the spacing between these repeated spectra decreases.
Aliasing, a phenomenon where overlapping of these repeated spectra occurs, can be avoided by ensuring that the spacing between the repeated spectra is greater than the signal’s bandwidth. This requires careful selection of the maximum value of T to maintain adequate separation between the repeated spectra.
In practice, filters are employed to isolate and pass frequencies within the desired range. These filters have specific constants and frequencies designed to allow only the frequencies within the narrow band to pass through, thus preserving the integrity of the original signal while preventing aliasing. This process ensures that the signal is accurately sampled and reconstructed without distortion.