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Digital signal processing (DSP) is the study of signals in a digital representation and the processing methods of these signals. DSP and analog signal processing are subsets of signal processing. It has three major subfields: audio signal processing, digital image processing and speech processing.

In DSP, engineers most commonly study digital signals in one of the following domains: time domain (one-dimensional signals), spatial domain (multidimensional signals), frequency domain, autocorrelation domain, and wavelet domains. They choose the domain in which to process a signal by making an educated guess (or trying out different possibilities) as to which domain best represents the essential characteristics of the signal. A sequence of samples from a measuring device produces a time or spatial domain representation, whereas a discrete Fourier transform produces the frequency domain information. The autocorrelation is, loosely speaking, defined as the expected value of correlation of the signal with itself on some distance in time or spatial distance.

## Signal sampling

A digital signal is often a numerical representation of a continuous signal. This discrete representation of a continuous signal will generally introduce some error in to the data. The accuracy of the representation is mostly dependent on two things; sampling frequency and the number of bits used for the representation. The continuous signal is usually sampled at regular intervals and the value of the continuous signal in that interval is represented by a discrete value. The sampling frequency or sampling rate is then the rate at which new samples are taken from the continuous signal. The number of bits used for one value of the discrete signal tells us how accurately the signal magnitude is represented. Similarly, the sampling frequency controls the temporal or spatial accuracy of the discrete signal.

The Nyquist-Shannon sampling theorem, a fundamental theorem of signal processing, states that a sampled signal cannot unambiguously represent signal components with frequencies above half the sampling frequency. This frequency (half the sampling frequency) is called the Nyquist frequency. Frequencies above the Nyquist frequency N can be observed in the digital signal, but their frequency is ambiguous. That is, a frequency component with frequency f cannot be distinguished from another component with frequency 2N-f, 2N+f, 4N-f, etc. This is called aliasing. To handle this problem as gracefully as possible, most analog signals are filtered with an anti-aliasing filter (usually a low-pass filter) at the Nyquist frequency before conversion to the digital representation.

## Time and spatial domains

The most common processing approach in the time or spatial domain is enhancement of the input signal through a method called filtering. Filtering consists generally of some transformation of a number of surrounding samples around the current sample of the input and/or output signal. Properties such as the following characterize filters:

• A "linear" filter consists of a linear transformation of input samples; other filters are "non-linear." Linear filters satisfy the superposition condition, i.e. if an input signal is a weighted linear combination of different input signals, the output will be an equally weighted linear combination of the corresponding individual output signals.

• A "causal" transformation uses only previous samples of the input or output signals; transformations that also use future input samples are "non- causal." Adding a delay will transform many non-causal filters into causal filters.

• A "time-invariant" filter has constant properties over time; other filters such as adaptive filters change in time.

• "Finite impulse response" (FIR) filters use only the input signal; so-called "infinite impulse response" filters use both the input signal and previous samples of the output signal.

Most filters can, in Z-domain (frequency domain is a subset of Z-domain), be described by their Transfer functions.

## Frequency domain

Signals are converted from time or spatial domain to the frequency domain usually through the Fourier transform. In Fourier transform the signal information is converted to a magnitude and phase component of each frequency. Regurarly, the Fourier transform is converted to the power spectrum, which is the magnitude of each frequency component squared. The most common purpose for analysis of signals in the frequency domain is analysis of signal properties. The engineer can study the spectrum to get information of which frequencies are present in the input signal and which are missing. However, there are some commonly used frequency domain transformations, for example, the cepstrum. In generation of the cepstrum, a signal is converted to the frequency domain through Fourier transform, then the logarithm is of the spectrum, which is converted back to time domain through the inverse Fourier transform. In the cepstrum, frequency components with smaller magnitude are thus emphasised while retaining the order of magnitudes of frequency components.

## Applications

Typical applications of digital signal processing are, for example, speech compression and transmission in (digital) mobile phones, equalisation of sound in Hifi-equipment, weather forecasting and economic forecasting, analysis and control of industrial processes, computer-generated animations in movies and image manipulation.

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