A digital delay line (or simply delay line, also called delay filter) is a discrete element in a digital filter, which allows a signal to be delayed by a number of samples. Delay lines are commonly used to delay audio signals feeding loudspeakers to compensate for the speed of sound in air, and to align video signals with accompanying audio, called audio-to-video synchronization. Delay lines may compensate for electronic processing latency so that multiple signals leave a device simultaneously despite having different pathways.
Digital delay lines are widely used building blocks in methods to simulate room acoustics, musical instruments and effects units. Digital waveguide synthesis shows how digital delay lines can be used as sound synthesis methods for various musical instruments such as string instruments and wind instruments.
If a delay line holds a non-integer value smaller than one, it results in a fractional delay line (also called interpolated delay line or fractional delay filter). A series of an integer delay line and a fractional delay filter is commonly used for modelling arbitrary delay filters in digital signal processing. [2] The Dattorro scheme is an industry standard implementation of digital filters using fractional delay lines. [3]
The standard delay line with integer delay is derived from the Z-transform of a discrete-time signal delayed by samples [4] :
In this case, is the integer delay filter with:
The discrete-time domain filter for integer delay as the inverse zeta transform of is trivial, since it is an impulse shifted by [5] :
Working in the discrete-time domain with fractional delays is less trivial. In its most general theoretical form, a delay line with arbitrary fractional delay is defined as a standard delay line with delay , which can be modelled as the sum of an integer component and a fractional component which is smaller than one sample:
| (Def. 1) |
This is the domain representation of a non-trivial digital filter design problem: the solution is an any time-domain filter that represents or approximates the inverse Z-transform of . [2]
The conceptually easiest solution is obtained by sampling the continuous-time domain solution, which is trivial for any delay value. Given a continuous-time signal delayed by samples, or seconds [6] :
In this case, is the continuous-time domain fractional delay filter with:
The naive solution for the sampled filter is the sampled inverse Fourier transform of , which produces a non-causal IIR filter shaped as a Cardinal Sine shifted by [6] :
The continuous-time domain is shifted by the fractional delay while the sampling is always aligned to the cartesian plane, therefore:
The conceptually easiest implementable solution is the causal truncation of the naive solution above. [7]
Truncating the impulse response might however cause instability, which can be mitigated in a few ways:
What follows is an expansion of the formula above displaying the resulting filters of order up to :
Lagrange Interpolator Formula Expansion [7] | ||||
---|---|---|---|---|
N = 1 | - | - | ||
N = 2 | - | |||
N = 3 |
Another approach is designing an IIR filter of order with a Z-transform structure that forces it to be an all-pass while still approximating a delay [7] :
The reciprocally placed zeros and poles of respectively flatten the frequency response, while the phase is function of the phase of . Therefore, the problem becomes designing the FIR filter , that is finding its coefficients as a function of D (note that always), so that the phase approximates best the desired value . [7]
The main solutions are:
What follows is an expansion of the formula above displaying the resulting coefficients of order up to :
Thiran All-Pole Low-Pass Filter Coefficients Formula Expansion [7] | ||||
---|---|---|---|---|
N = 1 | 1 | - | - | |
N = 2 | 1 | - | ||
N = 3 | 1 |
Digital delay lines were first used to compensate for the speed of sound in air in 1973 to provide appropriate delay times for the distant speaker towers at the Summer Jam at Watkins Glen rock festival in New York, with 600,000 people in the audience. New York City–based company Eventide Clock Works provided digital delay devices each capable of 200 milliseconds of delay. Four speaker towers were placed 200 feet (60 m) from the stage, their signal delayed 175 ms to compensate for the speed of sound between the main stage speakers and the delay towers. Six more speaker towers were placed 400 feet from the stage, requiring 350 ms of delay, and a further six towers were placed 600 feet away from the stage, fed with 525 ms of delay. Each Eventide DDL 1745 module contained one hundred 1000-bit shift register chips and a bespoke digital-to-analog converter, and cost $3,800 (equivalent to $26,585in 2022). [12] [13]
The Nyquist–Shannon sampling theorem is an essential principle for digital signal processing linking the frequency range of a signal and the sample rate required to avoid a type of distortion called aliasing. The theorem states that the sample rate must be at least twice the bandwidth of the signal to avoid aliasing distortion. In practice, it is used to select band-limiting filters to keep aliasing distortion below an acceptable amount when an analog signal is sampled or when sample rates are changed within a digital signal processing function.
A wavelet is a wave-like oscillation with an amplitude that begins at zero, increases or decreases, and then returns to zero one or more times. Wavelets are termed a "brief oscillation". A taxonomy of wavelets has been established, based on the number and direction of its pulses. Wavelets are imbued with specific properties that make them useful for signal processing.
A low-pass filter is a filter that passes signals with a frequency lower than a selected cutoff frequency and attenuates signals with frequencies higher than the cutoff frequency. The exact frequency response of the filter depends on the filter design. The filter is sometimes called a high-cut filter, or treble-cut filter in audio applications. A low-pass filter is the complement of a high-pass filter.
Chebyshev filters are analog or digital filters that have a steeper roll-off than Butterworth filters, and have either passband ripple or stopband ripple. Chebyshev filters have the property that they minimize the error between the idealized and the actual filter characteristic over the operating frequency range of the filter, but they achieve this with ripples in the passband. This type of filter is named after Pafnuty Chebyshev because its mathematical characteristics are derived from Chebyshev polynomials. Type I Chebyshev filters are usually referred to as "Chebyshev filters", while type II filters are usually called "inverse Chebyshev filters". Because of the passband ripple inherent in Chebyshev filters, filters with a smoother response in the passband but a more irregular response in the stopband are preferred for certain applications.
In mathematics, the Gibbs phenomenon is the oscillatory behavior of the Fourier series of a piecewise continuously differentiable periodic function around a jump discontinuity. The th partial Fourier series of the function produces large peaks around the jump which overshoot and undershoot the function values. As more sinusoids are used, this approximation error approaches a limit of about 9% of the jump, though the infinite Fourier series sum does eventually converge almost everywhere except points of discontinuity.
In signal processing, a finite impulse response (FIR) filter is a filter whose impulse response is of finite duration, because it settles to zero in finite time. This is in contrast to infinite impulse response (IIR) filters, which may have internal feedback and may continue to respond indefinitely.
The Butterworth filter is a type of signal processing filter designed to have a frequency response that is as flat as possible in the passband. It is also referred to as a maximally flat magnitude filter. It was first described in 1930 by the British engineer and physicist Stephen Butterworth in his paper entitled "On the Theory of Filter Amplifiers".
In mathematics and signal processing, the Hilbert transform is a specific singular integral that takes a function, u(t) of a real variable and produces another function of a real variable H(u)(t). The Hilbert transform is given by the Cauchy principal value of the convolution with the function (see § Definition). The Hilbert transform has a particularly simple representation in the frequency domain: It imparts a phase shift of ±90° (π⁄2 radians) to every frequency component of a function, the sign of the shift depending on the sign of the frequency (see § Relationship with the Fourier transform). The Hilbert transform is important in signal processing, where it is a component of the analytic representation of a real-valued signal u(t). The Hilbert transform was first introduced by David Hilbert in this setting, to solve a special case of the Riemann–Hilbert problem for analytic functions.
In signal processing, a comb filter is a filter implemented by adding a delayed version of a signal to itself, causing constructive and destructive interference. The frequency response of a comb filter consists of a series of regularly spaced notches in between regularly spaced peaks giving the appearance of a comb.
In mathematics, the discrete-time Fourier transform (DTFT), also called the finite Fourier transform, is a form of Fourier analysis that is applicable to a sequence of values.
The rectangular function is defined as
In signal processing, linear phase is a property of a filter where the phase response of the filter is a linear function of frequency. The result is that all frequency components of the input signal are shifted in time by the same constant amount, which is referred to as the group delay. Consequently, there is no phase distortion due to the time delay of frequencies relative to one another.
In system analysis, among other fields of study, a linear time-invariant (LTI) system is a system that produces an output signal from any input signal subject to the constraints of linearity and time-invariance; these terms are briefly defined below. These properties apply (exactly or approximately) to many important physical systems, in which case the response y(t) of the system to an arbitrary input x(t) can be found directly using convolution: y(t) = (x ∗ h)(t) where h(t) is called the system's impulse response and ∗ represents convolution (not to be confused with multiplication). What's more, there are systematic methods for solving any such system (determining h(t)), whereas systems not meeting both properties are generally more difficult (or impossible) to solve analytically. A good example of an LTI system is any electrical circuit consisting of resistors, capacitors, inductors and linear amplifiers.
Lanczos filtering and Lanczos resampling are two applications of a mathematical formula. It can be used as a low-pass filter or used to smoothly interpolate the value of a digital signal between its samples. In the latter case, it maps each sample of the given signal to a translated and scaled copy of the Lanczos kernel, which is a sinc function windowed by the central lobe of a second, longer, sinc function. The sum of these translated and scaled kernels is then evaluated at the desired points.
The zero-order hold (ZOH) is a mathematical model of the practical signal reconstruction done by a conventional digital-to-analog converter (DAC). That is, it describes the effect of converting a discrete-time signal to a continuous-time signal by holding each sample value for one sample interval. It has several applications in electrical communication.
First-order hold (FOH) is a mathematical model of the practical reconstruction of sampled signals that could be done by a conventional digital-to-analog converter (DAC) and an analog circuit called an integrator. For FOH, the signal is reconstructed as a piecewise linear approximation to the original signal that was sampled. A mathematical model such as FOH (or, more commonly, the zero-order hold) is necessary because, in the sampling and reconstruction theorem, a sequence of Dirac impulses, xs(t), representing the discrete samples, x(nT), is low-pass filtered to recover the original signal that was sampled, x(t). However, outputting a sequence of Dirac impulses is impractical. Devices can be implemented, using a conventional DAC and some linear analog circuitry, to reconstruct the piecewise linear output for either predictive or delayed FOH.
In functional analysis, the Shannon wavelet is a decomposition that is defined by signal analysis by ideal bandpass filters. Shannon wavelet may be either of real or complex type.
The spectrum of a chirp pulse describes its characteristics in terms of its frequency components. This frequency-domain representation is an alternative to the more familiar time-domain waveform, and the two versions are mathematically related by the Fourier transform.
The spectrum is of particular interest when pulses are subject to signal processing. For example, when a chirp pulse is compressed by its matched filter, the resulting waveform contains not only a main narrow pulse but, also, a variety of unwanted artifacts many of which are directly attributable to features in the chirp's spectral characteristics.
The simplest way to derive the spectrum of a chirp, now that computers are widely available, is to sample the time-domain waveform at a frequency well above the Nyquist limit and call up an FFT algorithm to obtain the desired result. As this approach was not an option for the early designers, they resorted to analytic analysis, where possible, or to graphical or approximation methods, otherwise. These early methods still remain helpful, however, as they give additional insight into the behavior and properties of chirps.
The chirp pulse compression process transforms a long duration frequency-coded pulse into a narrow pulse of greatly increased amplitude. It is a technique used in radar and sonar systems because it is a method whereby a narrow pulse with high peak power can be derived from a long duration pulse with low peak power. Furthermore, the process offers good range resolution because the half-power beam width of the compressed pulse is consistent with the system bandwidth.
Beamforming is a signal processing technique used to spatially select propagating waves. In order to implement beamforming on digital hardware the received signals need to be discretized. This introduces quantization error, perturbing the array pattern. For this reason, the sample rate must be generally much greater than the Nyquist rate.