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is the real part, and
and for a negative frequency, the phase, is decreasing with time (i.e. indicating rotation in the direction of negative angle, i.e. clockwise rotation).
For a signal to exhibit differences between positive and negative frequency components it must be complex, since real signals have Hermitian frequency spectra (i.e. if the signal is real, the real part of the spectrum must have even symmetry and the imaginary part must have odd symmetry).
Here is an example of a signal having a negative frequency:
The real part (shown by a solid line) is a cosine wave, and the imaginary part (shown by a dashed/dotted line) is a sine wave.
If we construct a parametric plot of real versus imaginary, we will observe that the signal traces a circle, as the parameter (time) evolves. The direction of rotation around the circle will be clockwise as time increases.
Since positive angle is defined as counterclockwise rotation, we have that the signal is moving in a direction of negative angle.
Since frequency is defined as the time derivative of phase, the constant speed of clockwise rotation indicates a constant negative frequency.
Sampling of positive and negative frequencies and aliasing
Here are some sampled frequencies to illustrate aliasing between positive and negative frequency when a waveform is sampled. The red indicates the lowest frequency (DC), then orange is next highest, yellow, green, blue, and violet, the highest frequency. "R" denotes real part (cosine) and "I" imaginary part (sine).
Note that, for example, a complex exponential (sine part shown) of fractional frequency +5/8 has the same samples as one of fractional frequency −3/8. Likewise +7/8 gives the same samples as −1/8. Similarly, fractional frequency of 1 (8 samples in 8 cycles) gives the same samples as DC (i.e. the samples are all ones).
Negative frequency as a matched filter for positive frequencies
The rows of the DFT matrix 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 filter 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 radar, the usual convention is that objects moving toward the radar are considered to induce a positive frequency, and objects going away are considered to induce a negative frequency.
If we make a three dimensional plot of the signal x + iy where , as a (x, y, t) plot, we can see more explicitly the clockwise rotation in a right hand coordinate system as time increases.
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