The concept of negative and positive frequency can be as simple as a wheel rotating one way or the other way. A
signed value of frequency indicates both the rate and direction of rotation. The rate is expressed in units such as revolutions (aka
cycles) per second (
hertzThe hertz is a unit of frequency. It is defined as the number of complete cycles per second. It is the basic unit of frequency in the International System of Units , and is used worldwide in both general-purpose and scientific contexts...
) or radians/second (where 1 cycle corresponds to 2π
radianThe radian is a unit of plane angle, equal to 180/π degrees, or about 57.2958 degrees, or about 57°17′45″. It is the standard unit of angular measurement in all areas of mathematics beyond the elementary level....
s).
A sinusoid is a function of an angular argument, and its amplitude varies cyclically as the
angle (aka
phaseThe phase of an oscillation or wave is the fraction of a complete cycle corresponding to an offset in the displacement from a specified reference point at time t = 0. Phase is a frequency domain or Fourier transform domain concept, and as such, can be readily understood in terms of simple harmonic...
) steadily increases or decreases.
The concept of negative and positive frequency can be as simple as a wheel rotating one way or the other way. A
signed value of frequency indicates both the rate and direction of rotation. The rate is expressed in units such as revolutions (aka
cycles) per second (
hertzThe hertz is a unit of frequency. It is defined as the number of complete cycles per second. It is the basic unit of frequency in the International System of Units , and is used worldwide in both general-purpose and scientific contexts...
) or radians/second (where 1 cycle corresponds to 2π
radianThe radian is a unit of plane angle, equal to 180/π degrees, or about 57.2958 degrees, or about 57°17′45″. It is the standard unit of angular measurement in all areas of mathematics beyond the elementary level....
s).
Sinusoids
A sinusoid is a function of an angular argument, and its amplitude varies cyclically as the
angle (aka
phaseThe phase of an oscillation or wave is the fraction of a complete cycle corresponding to an offset in the displacement from a specified reference point at time t = 0. Phase is a frequency domain or Fourier transform domain concept, and as such, can be readily understood in terms of simple harmonic...
) steadily increases or decreases. When the angle is a function of time, the concept of negative frequency is sometimes used to distinguish a decreasing angle from an increasing one. But sinusoids are not monotonic functions. Consequently, does not preserve the sign of , just as does not preserve the sign of . Note that represents a usually unknown, random phase offset. In most cases of dealing with a single, real-valued sinusoid, it is sufficient to
assume that is positive. It represents the frequency, in units of radians/sec.
Sometimes there are two sinusoids with the same frequency, and a known phase
difference, for instance
:
and
When , appears to
lead by cycle ( radians). But when , the roles are reversed. So in that case it is possible to distinguish negative and positive frequencies. The diagram depicts a negative frequency. and are referred to as
real and
imaginary, respectively. And .
A parametric plot of imaginary vs real would trace a circular path (like the rotating wheel). The addition of a time dimension creates a corkscrew pattern. A negative frequency (decreasing phase) causes a clockwise rotation in a right hand coordinate system as time increases
:
Complex sinusoids
The complex function
: facilitates many kinds of mathematical operations involving , due in large part to
Euler's simplificationEuler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that shows a deep relationship between the trigonometric functions and the complex exponential function...
:
This very useful form is often referred to as a
complex sinusoid, and it preserves the distinction between positive and negative .
- For positive values, it is also called the analytic representation
In mathematics and signal processing, the analytic representation of a real-valued function or signal facilitates many mathematical manipulations of the signal. The basic idea is that the negative frequency components of the Fourier transform of a real-valued function are superfluous, due to the...
of .
The
Fourier transformIn mathematics, Fourier analysis is a subject area which grew out of the study of Fourier series. The subject began with trying to understand when it was possible to represent general functions by sums of simpler trigonometric functions...
of produces a non-zero response only at frequency .
- The transform of has responses at both and , which reflects the fact that is insufficient to determine the sign of .
- So just as is either +2 or -2, the interpretation of the ambiguity often depends on collateral information.
- An alternative, and surprisingly useful, viewpoint is that both frequencies are present, as implied by the inverse of Euler's formula
Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that shows a deep relationship between the trigonometric functions and the complex exponential function...
: .
Sampling of positive and negative frequencies and aliasing
When a complex sinusoid is sampled at regular intervals, its frequency becomes indistinguishable from certain other frequencies, including negative ones (referred to as
aliasingIn signal processing and related disciplines, aliasing refers to an effect that causes different signals to become indistinguishable when sampled...
). The adjacent figure illustrates this effect for several cases. The red indicates 0 Hz (aka
DC). Successively higher frequencies are indicated by orange, blue, purple, violet, black, and blue. Note that some frames depict "R" and "I" for the same frequency, and others depict the "I" samples of different frequencies that are aliases of each other.
For instance, the fourth frame (purple and green) compares samples of the imaginary component of the fractional frequency + with those of negative frequency , to illustrate that they are indistinguishable. Or in other words
: for integer values of n, representing the sample number. The underlying waveforms are just the imaginary components of
: and , where is the sample rate (samples/sec).
Likewise + is indistinguishable from . And (last plot) is indistinguishable from (first plot).
Negative frequency as a matched filter for positive frequencies
The rows of the
DFT matrixA DFT matrix is an expression of a discrete Fourier transform as a matrix multiplication.-Definition:An N-point DFT is expressed as an N-by-N matrix multiplication as , where is the original input signal, and is the DFT of the signal.The transformation of size can be defined as , or...
begin at zero frequency, and get more negative as we move downward, row by row. This is because each of these rows functions as a
matched filterIn telecommunications, a matched filter is obtained by correlating a known signal, or template, with an unknown signal to detect the presence of the template in the unknown signal. This is equivalent to convolving the unknown signal with a conjugated time-reversed version of the template...
to measure increasingly positive frequencies in the signal under test. For example, the top row of the 8 point DFT matrix measures DC in the signal, while the next row, which is a signal of fractional frequency −1/8, measures the strength at +1/8 fractional frequency in the signal under test.
Negative frequency in Doppler radar
In
Doppler radarDoppler radar is radar that makes use of the doppler effect to produce data about objects at a distance. It does this by beaming a microwave signal towards a desired target and listening for its reflection, then analyzing how the original signal has been altered by the object that reflected it...
, the usual convention is that objects moving toward the radar are considered to induce a positive (differential) frequency, and objects going away are considered to induce a negative frequency.
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