In
digital signal processing
Digital signal processing (DSP) is the use of digital processing, such as by computers or more specialized digital signal processors, to perform a wide variety of signal processing operations. The digital signals processed in this manner are ...
(DSP), a normalized frequency () is a
quantity
Quantity or amount is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value multiple of a u ...
that is equal to the ratio of a
frequency
Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
and a characteristic frequency of a system.
An example of a normalized frequency is the sampling frequency in a system in which a signal is sampled at periodically, in which it equals (with the unit ''cycle per sample''), where is a frequency and is the ''
sampling rate
In signal processing, sampling is the reduction of a continuous-time signal
In mathematical dynamics, discrete time and continuous time are two alternative frameworks within which variables that evolve over time are modeled.
Discrete time
...
''. For regularly spaced sampling, the
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous g ...
time variable, (with unit
second
The second (symbol: s) is the unit of time in the International System of Units (SI), historically defined as of a day – this factor derived from the division of the day first into 24 hours, then to 60 minutes and finally to 60 seconds ea ...
), is replaced by a
discrete ''sampling
count
Count (feminine: countess) is a historical title of nobility in certain European countries, varying in relative status, generally of middling rank in the hierarchy of nobility. Pine, L. G. ''Titles: How the King Became His Majesty''. New York ...
'' variable, (with the unit sample), upon division by the sampling interval, (with the unit second per sample).
The use of normalized frequency allows us to present concepts that are universal to all sample rates in a way that is independent of the sample rate. An example of such a concept is a digital filter design whose bandwidth is specified not in
hertz
The hertz (symbol: Hz) is the unit of frequency in the International System of Units (SI), equivalent to one event (or cycle) per second. The hertz is an SI derived unit whose expression in terms of SI base units is s−1, meaning that o ...
, but as a percentage of the sample rate of the data passing through it. Formulas expressed in terms of or 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, , with or .
Alternative normalizations
Some programs (such as
MATLAB
MATLAB (an abbreviation of "MATrix LABoratory") is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks. MATLAB allows matrix manipulations, plotting of functions and data, implementat ...
) that design filters with real-valued coefficients use the Nyquist frequency () as the
normalization constant
The concept of a normalizing constant arises in probability theory and a variety of other areas of mathematics. The normalizing constant is used to reduce any probability function to a probability density function with total probability of one.
...
. The resultant normalized frequency has units of ''half-cycles/sample'' or equivalently ''cycles per 2 samples''.
Angular frequency
In physics, angular frequency "''ω''" (also referred to by the terms angular speed, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit ti ...
, denoted by and with the unit
radian per second
The radian per second (symbol: rad⋅s−1 or rad/s) is the unit of angular velocity in the International System of Units (SI). The radian per second is also the SI unit of angular frequency, commonly denoted by the Greek letter ''ω'' (omeg ...
, can be similarly normalized. When is normalized with reference to the sampling rate, the resulting unit is radian per sample. The normalized Nyquist angular frequency is ''π'' radians/sample.
The following table shows examples of normalized frequencies for a 1 kHz signal, a sampling rate =
44.1 kHz
In digital audio, 44,100 Hz (alternately represented as 44.1 kHz) is a common sampling frequency. Analog audio is often recorded by sampling it 44,100 times per second, and then these samples are used to reconstruct the audio signal w ...
, and 3 different choices of normalized units. Also shown is the frequency region containing one cycle of the
discrete-time Fourier transform
In mathematics, the discrete-time Fourier transform (DTFT) is a form of Fourier analysis that is applicable to a sequence of values.
The DTFT is often used to analyze samples of a continuous function. The term ''discrete-time'' refers to the ...
, which is always a periodic function.
See also
*
Prototype filter
Notes and citations
{{DEFAULTSORT:Normalized Frequency (Digital Signal Processing)
Digital signal processing
Frequency