Discrete-time processing of continuous signals
Introduction
There are several advantages to using the discrete-time system to process continuous signals. The advantages result from the power and flexibility of discrete-time computing devices. It is easier to perform a broad class of signal manipulations using the arithmetic operations of a computer than through the use of analog components. Also, implementing a system in a computer only involves writing a set of instructions or programs for the computer to execute. Then, the discrete-time system is easily changed by modifying the computer program. Often the system can be modified in real-time to optimize some criterion associated with the processed signal. Another advantage of discrete-time processing is that the direct dependence of the dynamic range and signal to noise ratio on the number of bits used to represent the discrete-time signal. These advantages have led to a proliferation of computing devices designed specifically for discrete-time signal processing.
A minimal system for the discrete-time processing of continuous-time signals must contain a sampling device, as well as a computing device for implementing the discrete-time system. In addition to that, if the processed signal is to be converted back to continuous time, then the reconstruction is necessary. Decimation and interpolation are methods for changing the effective sampling rate, while interpolation increases the effective sampling rate. Apt use of the basic system for processing continuous-time signals. It is concluded by revisiting oversampling and examining the role of interpolation and decimation in systems that process continuous-time signals.
A basic discrete-time signal processing system
In a basic discrete-time signal processing system, firstly a continuous-time signal is passed through a low pass anti-aliasing filter and then sampled at the intervals of T. It is done to convert it to a discrete-time signal. The sampled signal is then processed by a discrete-time system to impart some desired effect to the signal. For example, the discrete-time system may represent a filter designed to have a specific frequency response, like an equalizer. After processing, the signal is then converted back to a continuous-time format. A zero-order hold device converts the discrete-time signal back to continuous time, and an anti-imaging filter removes the distortion introduced by the zero-order hold.
This combination can be reduced to an equivalent continuous-time filter by using the FT as an analysis tool. The idea is to find a continuous-time system: g(t) ←→ G(jw) such that Y(jw) = G(jw) X(jw). Therefore, G(jw) has the same effect on the system as the output.
Oversampling
An anti-aliasing filter prevents aliasing by limiting the bandwidth of the signal prior to sampling. While the signal of interest may have maximum frequency W, in general, the continuous-time signal will have energy at higher frequencies due to the presence of noise and other non-essential characteristics. The anti-aliasing filter is chosen to prevent the noise from moving back down into the band of interest and producing aliases there. The magnitude response of practical anti-aliasing filter will not go from unit gain to zero at frequency W, but instead goes from passband to stopband over a range of frequencies.
Also, it can be noted that the transition band of the anti-aliasing filter must be less than the sampling frequency minus twice the frequency of the highest frequency component of interest in the signal. Filter with small transition bands is difficult to design and expensive. By oversampling, we can greatly relax the requirements on the anti-aliasing filter transition band, and consequently, reduce its complexity and cost.
In both, the sampling and the reconstruction, the difficulties of implementing practical analog filters suggest using the highest possible sampling rate. Though, if the data set in sampling rates lead to increased discrete-time system cost because the discrete-time system should perform its computation at a much faster rate. This conflict over the sampling rate is mitigated if we are able to change the sampling rate such that a high rate is used for sampling and reconstruction and a lower rate is used for discrete-time processing. Decimation and interpolation offer that, we will be discussing them next.
Decimation
Let us consider the DTFTs obtained by sampling an identical continuous-time signal at different intervals, Ts1 and T s2 . Let the sampled signals be denoted as x1[n] and x2[n]. Then we assume Ts1 = qT s2 , where q is an integer, and that aliasing does not occur at either sampling rate. Decimation corresponds to changing X2 (e^iΩ) to X1 (e^iΩ). One way to do this is to convert the discrete-time sequence back to a continuous-time signal and then resample. This approach is subject to distortion introduced in the reconstruction operation. Distortion can be avoided by using methods that operate directly on the discrete-time signals to change the sampling rate.
Subsampling is the key to reducing the sampling rate.
Interpolation
Interpolation increases the sampling rate and requires us to produce values between the samples of the signal. The interpolation factor q is accomplished by inserting q-1 zeros in between each sample of x1[n] and then low-pass filtering. It is also known as upsampling and is often denoted by an upwards arrow followed by the interpolation factor.
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