Group Delay

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Index: Introduction to Digital Filters with Audio Applications

Introduction to Digital Filters with Audio Applications

Phase and Group Delay

Phase Unwrapping

Derivation of Group Delay as Modulation Delay

The amplitude envelope, or relatively slowly-changing outline of a sound waveform, makes for a useful first approximation of the instantaneous loudness of the sound. — Click for http://en.wikipedia.org/wiki/ADSR_envelope

The size of the passband, typically expressed in Hz. — Click for http://en.wikipedia.org/wiki/Bandpass

A sinusoid is any function of the form A sin(ω t+φ), where t is the independent variable, and A, ω, φ are fixed parameters of the sinusoid called the amplitude, (radian) frequency, and phase, respectively. Sinusoidal motion is produced by any 'pure' vibration, such as that of an ideal tuning fork or mass-spring system. — Click for https://ccrma.stanford.edu/~jos/mdft/Sinusoids.html

Click for https://en.wikipedia.org/wiki/Envelope_(waves)

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A lowpass filter rejects high frequencies and does not affect low frequencies. — Click for http://ccrma.stanford.edu/~jos/filters/Simplest_Lowpass_Filter_I.html

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Transients are short intervals during which the signal evolves quickly in some nontrivial or relatively unpredictable way (Bello, 2005). In computer music, the short attack portion of a sound is often treated as a transient signal. — Click for http://en.wikipedia.org/wiki/Transient_%28acoustics%29

A signal is typically a real-valued function of time. A discrete-time signal is typically a real-valued function of discrete time, and is therefore a time-ordered sequence of real numbers. — Click for http://ccrma.stanford.edu/~jos/filters/Definition_Signal.html

Any function that is not linear is nonlinear. There are no general methods for solving nonlinear equations or nonlinear differential equations. This is why it is usually easier, although generally less accurate, to approximate a nonlinear function by a linear function if possible. — Click for http://en.wikipedia.org/wiki/Nonlinear

The phase delay of an LTI filter is minus the phase response divided by frequency — Click for https://ccrma.stanford.edu/~jos/filters/Phase_Delay.html

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The phase response of an LTI filter is the angle of the (complex) frequency response — Click for https://ccrma.stanford.edu/~jos/filters/Phase_Response_I_I.html

A filter in the audio signal processing context is any operation that accepts a signal as an input and produces a signal as an output. Most practical audio filters are linear and time invariant, in which case they can be characterized by their impulse response or their frequency response. — Click for https://ccrma.stanford.edu/~jos/filters/What_Filter.html

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JOS Home Page

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Index of terms in JOS Website

Index: Introduction to Digital Filters with Audio Applications

Introduction to Digital Filters with Audio Applications

Phase and Group Delay

Phase Unwrapping

Derivation of Group Delay as Modulation Delay

Group Delay

A more commonly encountered representation of filter phase response is called the group delay, defined by

$\displaystyle \zbox {D(\omega) \isdefs - \frac{d}{d\omega} \Theta(\omega).} \qquad\hbox{(Group Delay)}$

For linear phase responses, i.e., $\Theta(\omega) = -\alpha\omega$ for some constant $\alpha$ , the group delay and the phase delay are identical, and each may be interpreted as time delay (equal to $\alpha$ samples when $\omega\in[-\pi,\pi]$ ). If the phase response is nonlinear, then the relative phases of the sinusoidal signal components are generally altered by the filter. A nonlinear phase response normally causes a ``smearing'' of attack transients such as in percussive sounds. Another term for this type of phase distortion is phase dispersion. This can be seen below in §7.6.5.

An example of a linear phase response is that of the simplest lowpass filter, $\Theta(\omega) = -\omega T/2 \,\,\Rightarrow\,\, P(\omega)=D(\omega)=T/2$ . Thus, both the phase delay and the group delay of the simplest lowpass filter are equal to half a sample at every frequency.

For any reasonably smooth phase function, the group delay $D(\omega)$ may be interpreted as the time delay of the amplitude envelope of a sinusoid at frequency $\omega$ [63]. The bandwidth of the amplitude envelope in this interpretation must be restricted to a frequency interval over which the phase response is approximately linear. We derive this result in the next subsection.

Thus, the name ``group delay'' for $D(\omega)$ refers to the fact that it specifies the delay experienced by a narrow-band ``group'' of sinusoidal components which have frequencies within a narrow frequency interval about $\omega$ . The width of this interval is limited to that over which $D(\omega)$ is approximately constant.

Subsections

Derivation of Group Delay as Modulation Delay

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``Introduction to Digital Filters with Audio Applications'', by Julius O. Smith III, (September 2007 Edition)
Copyright © 2024-09-03 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

Group Delay

``Introduction to Digital Filters with Audio Applications'', by Julius O. Smith III, (September 2007 Edition) Copyright © 2024-09-03 by Julius O. Smith III Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

``Introduction to Digital Filters with Audio Applications'', by Julius O. Smith III, (September 2007 Edition)
Copyright © 2024-09-03 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA), Stanford University