- Source: Normalized frequency (signal processing)
In digital signal processing (DSP), a normalized frequency is a ratio of a variable frequency (
f
{\displaystyle f}
) and a constant frequency associated with a system (such as a sampling rate,
f
s
{\displaystyle f_{s}}
). Some software applications require normalized inputs and produce normalized outputs, which can be re-scaled to physical units when necessary. Mathematical derivations are usually done in normalized units, relevant to a wide range of applications.
Examples of normalization
A typical choice of characteristic frequency is the sampling rate (
f
s
{\displaystyle f_{s}}
) that is used to create the digital signal from a continuous one. The normalized quantity,
f
′
=
f
f
s
,
{\displaystyle f'={\tfrac {f}{f_{s}}},}
has the unit cycle per sample regardless of whether the original signal is a function of time or distance. For example, when
f
{\displaystyle f}
is expressed in Hz (cycles per second),
f
s
{\displaystyle f_{s}}
is expressed in samples per second.
Some programs (such as MATLAB toolboxes) that design filters with real-valued coefficients prefer the Nyquist frequency
(
f
s
/
2
)
{\displaystyle (f_{s}/2)}
as the frequency reference, which changes the numeric range that represents frequencies of interest from
[
0
,
1
2
]
{\displaystyle \left[0,{\tfrac {1}{2}}\right]}
cycle/sample to
[
0
,
1
]
{\displaystyle [0,1]}
half-cycle/sample. Therefore, the normalized frequency unit is important when converting normalized results into physical units.
A common practice is to sample the frequency spectrum of the sampled data at frequency intervals of
f
s
N
,
{\displaystyle {\tfrac {f_{s}}{N}},}
for some arbitrary integer
N
{\displaystyle N}
(see § Sampling the DTFT). The samples (sometimes called frequency bins) are numbered consecutively, corresponding to a frequency normalization by
f
s
N
.
{\displaystyle {\tfrac {f_{s}}{N}}.}
: p.56 eq.(16) The normalized Nyquist frequency is
N
2
{\displaystyle {\tfrac {N}{2}}}
with the unit 1/Nth cycle/sample.
Angular frequency, denoted by
ω
{\displaystyle \omega }
and with the unit radians per second, can be similarly normalized. When
ω
{\displaystyle \omega }
is normalized with reference to the sampling rate as
ω
′
=
ω
f
s
,
{\displaystyle \omega '={\tfrac {\omega }{f_{s}}},}
the normalized Nyquist angular frequency is π radians/sample.
The following table shows examples of normalized frequency for
f
=
1
{\displaystyle f=1}
kHz,
f
s
=
44100
{\displaystyle f_{s}=44100}
samples/second (often denoted by 44.1 kHz), and 4 normalization conventions:
See also
Prototype filter
References
Kata Kunci Pencarian:
- Normalized frequency (signal processing)
- Normalized frequency
- Normalization
- Sampling (signal processing)
- Spectral density
- Coherence (signal processing)
- Hertz
- Signal-to-noise ratio
- Discrete time and continuous time
- Sonar signal processing
