**NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW
VERSION:** "Mathematics of the Discrete
Fourier Transform (DFT), with Audio Applications --- Second
Edition", by Julius
O. Smith III, W3K
Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright ©
*2017-09-28* by Julius O. Smith III -
Center for Computer Research
in Music and Acoustics (CCRMA), Stanford University

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## Frequency Response

Beginning with Eq. (B.4.3), we have

where is the transform of the filter input, is the transform of the output signal, and is the filter transfer function.

Definition.Thefrequency responseof a linear time-invariant digital filter is defined to be the transfer function, , evaluated on the unit circle, that is, .The frequency response is a complex-valued function of a real variable. The response at frequency Hz, for example, is , where is the sampling period in seconds.

Since every complex number can be represented as a magnitude and angle, the frequency response may be decomposed into two real-valued functions, the

amplitude responseand thephase response. Formally, we may define them as follows:

so that

Thus is the magnitude (or complex modulus) of , and is the phase (or complex angle) of .

Definition.The real valued function in Eq. (B.21) is called the filteramplitude responseand it specifies the amplitudegainthat the filter provides at each frequency.

Definition.The function is called thepower responseand it specifies the power gain at each frequency.

Definition.The real function in Eq. (B.21) is thephase responseand it gives the phase shift in radians that each input component sinusoid will undergo.If the filter input and output signals are and respectively, then

(B.22) (B.23)