Main article: Sampling signal processing To digitally analyze and manipulate an analog signal, it must be digitized with an analog-to-digital converter ADC. Discretization means that the signal is divided into equal intervals of time, and each interval is represented by a single measurement of amplitude. Quantization means each amplitude measurement is approximated by a value from a finite set. Rounding real numbers to integers is an example. The Nyquist—Shannon sampling theorem states that a signal can be exactly reconstructed from its samples if the sampling frequency is greater than twice the highest frequency component in the signal. In practice, the sampling frequency is often significantly higher than twice the Nyquist frequency.

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Main article: Sampling signal processing To digitally analyze and manipulate an analog signal, it must be digitized with an analog-to-digital converter ADC. Discretization means that the signal is divided into equal intervals of time, and each interval is represented by a single measurement of amplitude.

Quantization means each amplitude measurement is approximated by a value from a finite set. Rounding real numbers to integers is an example. The Nyquist—Shannon sampling theorem states that a signal can be exactly reconstructed from its samples if the sampling frequency is greater than twice the highest frequency component in the signal.

In practice, the sampling frequency is often significantly higher than twice the Nyquist frequency. Numerical methods require a quantized signal, such as those produced by an ADC. The processed result might be a frequency spectrum or a set of statistics. But often it is another quantized signal that is converted back to analog form by a digital-to-analog converter DAC.

Domains Edit In DSP, engineers usually study digital signals in one of the following domains: time domain one-dimensional signals , spatial domain multidimensional signals , frequency domain , and wavelet domains.

They choose the domain in which to process a signal by making an informed assumption or by trying different possibilities as to which domain best represents the essential characteristics of the signal and the processing to be applied to it. A sequence of samples from a measuring device produces a temporal or spatial domain representation, whereas a discrete Fourier transform produces the frequency domain representation.

Time and space domains Edit Main article: Time domain The most common processing approach in the time or space domain is enhancement of the input signal through a method called filtering. Digital filtering generally consists of some linear transformation of a number of surrounding samples around the current sample of the input or output signal. There are various ways to characterize filters; for example: A linear filter is a linear transformation of input samples; other filters are nonlinear.

Linear filters satisfy the superposition principle , i. A causal filter uses only previous samples of the input or output signals; while a non-causal filter uses future input samples.

A non-causal filter can usually be changed into a causal filter by adding a delay to it. A time-invariant filter has constant properties over time; other filters such as adaptive filters change in time.

A stable filter produces an output that converges to a constant value with time, or remains bounded within a finite interval. An unstable filter can produce an output that grows without bounds, with bounded or even zero input. A finite impulse response FIR filter uses only the input signals, while an infinite impulse response IIR filter uses both the input signal and previous samples of the output signal. A filter can be represented by a block diagram , which can then be used to derive a sample processing algorithm to implement the filter with hardware instructions.

A filter may also be described as a difference equation , a collection of zeros and poles or an impulse response or step response. The output of a linear digital filter to any given input may be calculated by convolving the input signal with the impulse response.

Main article: Frequency domain Signals are converted from time or space domain to the frequency domain usually through use of the Fourier transform. The Fourier transform converts the time or space information to a magnitude and phase component of each frequency.

With some applications, how the phase varies with frequency can be a significant consideration. Where phase is unimportant, often 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 determine which frequencies are present in the input signal and which are missing. Frequency domain analysis is also called spectrum- or spectral analysis.

Filtering, particularly in non-realtime work can also be achieved in the frequency domain, applying the filter and then converting back to the time domain. This can be an efficient implementation and can give essentially any filter response including excellent approximations to brickwall filters. There are some commonly used frequency domain transformations.

For example, the cepstrum converts a signal to the frequency domain through Fourier transform, takes the logarithm, then applies another Fourier transform. This emphasizes the harmonic structure of the original spectrum.

The Z-transform provides a tool for analyzing stability issues of digital IIR filters. It is analogous to the Laplace transform , which is used to design and analyze analog IIR filters. The original image is high-pass filtered, yielding the three large images, each describing local changes in brightness details in the original image.

It is then low-pass filtered and downscaled, yielding an approximation image; this image is high-pass filtered to produce the three smaller detail images, and low-pass filtered to produce the final approximation image in the upper-left.

In numerical analysis and functional analysis , a discrete wavelet transform is any wavelet transform for which the wavelets are discretely sampled. As with other wavelet transforms, a key advantage it has over Fourier transforms is temporal resolution: it captures both frequency and location information.

The accuracy of the joint time-frequency resolution is limited by the uncertainty principle of time-frequency.

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