Digital Signal Processing/Sampling and Reconstruction

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digital pro[edit]

Sampling is the process of recording the values of a signal at given points in time. For A/D converters, these points in time are equidistant. The number of samples taken during one second is called the sample rate. Keep in mind that these samples are still analogue values. The mathematic description of the ideal sampling is the multiplication of the signal with a sequence of dirac pulses. In real A/D converters the sampling is carried out by a sample-and-hold buffer. The sample-and-hold buffer splits the sample period in a sample time and a hold time. In case of a voltage being sampled, a capacitor is switched to the input line during the sample time. During the hold time it is detached from the line and keeps its voltage.

Analog Waveform.svg
Sampled Waveform.svg

Quantization[edit]

Quantization is the process of representing the analog voltage from the sample-and-hold circuit by a fixed number of bits. The input analog voltage (or current) is compared to a set of pre-defined voltage (or current) levels. Each of the levels is represented by a unique binary number, and the binary number that corresponds to the level that is closest to the analog voltage is chosen to represent that sample. This process rounds the analog voltage to the nearest level, which means that the digital representation is an approximation to the analog voltage. There are a few methods for quantizing samples. The most commonly used ones are the dual slope method and the successive approximation.

Digital Waveform.svg

Dual Slope[edit]

Successive Approximation[edit]

Reconstruction[edit]

Reconstruction is the process of creating an analog voltage (or current) from samples. A digital-to-analog converter takes a series of binary numbers and recreates the voltage (or current) levels that corresponds to that binary number. Then this signal is filtered by a lowpass filter. This process is analogous to interpolating between points on a graph, but it can be shown that under certain conditions the original analog signal can be reconstructed exactly from its samples. Unfortunately, the conditions for exact reconstruction cannot be achieved in practice, and so in practice the reconstruction is an approximation to the original analog signal.

Reconstructed Waveform.svg

Aliasing[edit]

Aliasing is a common problem in digital media processing applications. Many readers have heard of "anti-aliasing" features in high-quality video cards. This page will explain what Aliasing is, and how it can by avoided.

Aliasing is an effect of violating the Nyquist-Shannon sampling theory. During sampling the base band spectrum of the sampled signal is mirrored to every multifold of the sampling frequency. These mirrored spectra are called alias. If the signal spectrum reaches farther than half the sampling frequency base band spectrum and aliases touch each other and the base band spectrum gets superimposed by the first alias spectrum. The easiest way to prevent aliasing is the application of a steep sloped low-pass filter with half the sampling frequency before the conversion. Aliasing can be avoided by keeping Fs>2Fmax.

Nyquist Sampling Rate[edit]

The Nyquist Sampling Rate is the lowest sampling rate that can be used without having aliasing. The sampling rate for an analog signal must be at least two times the bandwidth of the signal. So, for example, an audio signal with a bandwidth of 20 kHz must be sampled at least at 40 kHz to avoid aliasing. In audio CD's, the sampling rate is 44.1 kHz, which is about 10% higher than the Nyquist Sampling Rate to allow cheaper reconstruction filters to be used.

Anti-Aliasing[edit]

The sampling rate for an analog signal must be at least two times as high as the highest frequency in the analog signal in order to avoid aliasing. Conversely, for a fixed sampling rate, the highest frequency in the analog signal can be no higher than one half of the sampling rate. Any part of the signal or noise that is higher than one half of the sampling rate will cause aliasing. In order to avoid this problem, the analog signal is usually filtered by a lowpass filter prior to being sampled, and this filter is called an anti-aliasing filter. Sometimes the reconstruction filter after a digital-to-analog converter is also called an anti-aliasing filter.

Converters[edit]

As a matter of professional interest, we will use this page to briefly discuss Analog-to-Digital (A2D) and Digital-to-Analog (D2A) converters, and how they are related to the field of digital signal processing. Strictly speaking, this page is not important to the core field of DSP, so it can be skipped at the reader's leisure.

A2D converters are the bread and butter of DSP systems. A2D converters change analog electrical data into digital signals through a process called sampling.

D2A converters attempt to create an analog output waveform from a digital signal. However, it is nearly impossible to create a smooth output curve in this manner, so the output waveform is going to have a wide bandwidth, and will have jagged edges. Some techniques, such as interpolation can be used to improve these output waveforms.