Name: sampling narrow frequency bands
Uploaded: 2024-09-26T13:17:07.000Z
Duration: 1 min 17 s
Channel: JoVE

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&#39;s spectrum has a lower frequency limit,&#160; &#969;1, &#8203;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&#39;s spectrum with the impulse train spectrum over 2&#8203;&#960;.
This convolution causes the signal&#39;s spectrum to repeat periodically with a period of 2&#960;/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&#39;s bandwidth and with the maximum value of T.&#160;
Filters with specific constants and frequencies pass the frequencies within the desired range.

bandpass sampling

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&#39;s bandwidth to prevent aliasing.
A bandpass signal has a spectrum with a lower frequency limit, denoted as &#969;1, and an upper frequency limit, denoted as &#969;2. 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&#39;s original spectrum with the spectrum of the impulse train over a range of 2&#960;. This convolution process causes the signal&#8217;s spectrum to repeat periodically, with a period of 2&#960;/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&#8217;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.

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 ...

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Unit 1

Sampling

KEY TERMS AND CONCEPTS

sampling narrow frequency bands

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 creates a sampled signal with repeating spectra.

The sampled signal spectrum is found by convolving the original spectrum with the spectrum of the impulse train over a range of 2π. That convolution makes the spectrum repeat periodically, with a period of 2π/T, where T is the sampling interval. As T increases, the repeated spectra move closer together.

Aliasing happens when those repeated spectra overlap. To avoid it, the spacing between the repeated spectra must stay greater than the signal bandwidth. That means the maximum value of T has to be chosen carefully so the spectra remain separated.

In practice, filters help isolate the desired narrow band. These filters use specific constants and frequencies so only the needed frequencies pass through. This keeps the original signal intact and allows accurate sampling and reconstruction without distortion.

Sampling Narrow Frequency Bands

sampling theorem

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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...

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Impulse-Train Sampling and Zero-Order Hold

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...

reconstruction of signal using interpolation

reconstruction-of-signal-using-interpolation

Signal Reconstruction with Low-Pass Filtering

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...

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Sampling Signals Without Aliasing

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...

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Decimation and Sampling Rate Reduction

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...

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Zero-Inserted Signal Reconstruction

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...

key terms and concepts

Learn how bandpass sampling captures signals in a narrow frequency band, avoids aliasing, and uses filters to preserve the original signal.

Sampling Narrow Frequency Bands | JoVE K12

JoVE Core Electrical Engineering offers easy-to-understand animated videos that cover fundamental concepts of electric circuits along with circuit analysis techniques. It incorporates first-order and second-order circuits using operational amplifiers and circuit transformation concepts. The inclusion of practical design examples and real-world applications through scientist-in-action videos adds significant educational value, making complex concepts accessible.

electrical engineering

Sampling Narrow Frequency Bands

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