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Normalized frequency (digital signal processing)

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Title: Normalized frequency (digital signal processing)  
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Subject: Frequency, Hertz, Normalized frequency, Sampling rate, Cycle per second, Radian per second
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Normalized frequency (digital signal processing)

In digital signal processing, the normalized frequency of a periodic signal is its frequency expressed in units of cycles (or radians) per sample, rather than in the usual SI units of hertz (cycles per second). The cycles-per-sample frequency is computed by dividing the cycles-per-second frequency by the sampling rate (samples per second); symbolically, the "per second" units cancel:

(cycles/second) / (samples/second) = cycles/sample.

Formulas expressed in terms of the sampling rate, \scriptstyle f_s,  and/or the sampling interval, \scriptstyle T = \frac{1}{f_s},  are readily converted to normalized frequency by setting those parameters to 1. The inverse operation is usually accomplished by replacing instances of the frequency parameter, \scriptstyle f ,  with \scriptstyle f/f_s  or  \scriptstyle f\cdot T.

Applications

The abstract reason for using normalized frequency is that, from the point of view of signal processing, a second is an arbitrary unit of time, while the sampling interval is a meaningful quantity (formally, a characteristic unit for the system): the frequency of a signal with respect to 1 second does not tell you about the behavior of the signal, but the frequency of a signal with respect to the sampling interval tells you the effect of sampling on the signal, via the sampling theorem. Stated alternatively, this process is called "normalization", and the sampling frequency is a normalizing constant.

In filter design, a given design can be used at different sample-rates, resulting in different frequency responses. Normalization produces a distribution that is independent of the sample rate, and thus one plot is sufficient for all possible sample rates.

Alternative normalizations

The reference value is usually the sampling frequency, denoted \scriptstyle f_\mathrm{s},\,  in samples per second, because the frequency spectrum of a sampled signal (with real or complex values) is periodic with period \scriptstyle f_\mathrm{s}.\,  When the actual frequency \scriptstyle ,\ f,  has units of hertz (SI units), the normalized frequencies, also denoted by \scriptstyle f,  have units of cycles per sample, and the periodicity of the normalized spectrum is 1.

Alternatively, if the actual frequency \scriptstyle ,\ \omega,  is written with units of radians per second (angular frequency), the normalized frequencies have units of radians per sample, and the periodicity of the distribution is 2π.

If a sampled waveform is real-valued, such as a typical filter impulse response, the periodicity of the frequency distribution is still \scriptstyle f_\mathrm{s}.  But due to symmetry, it is completely defined by the content within a span of just \scriptstyle f_\mathrm{s}/2, half the sampling frequency – the Nyquist frequency.  Accordingly, some filter design procedures/applications use that as the normalization reference (and the resulting units are half-cycles per sample).

Example

The following table shows examples of normalized frequencies for a 1 kHz signal, a sample rate \scriptstyle f_\mathrm{s} = 44.1 kHz, and these 3 different choices of normalization constant

Type Computation Value
Radians/sample 2 π 1000 / 44100 0.1425
cycles/sample
(w.r.t. fs, sampling frequency)
1000 / 44100 0.02268
half-cycles/sample
(w.r.t. fs/2, Nyquist frequency)
1000 / 22050 0.04535

See also

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