1. 8.6 Autocorrelation
instrument, mathematical definition, and properties
autocorrelation and Fourier transforms
cosine and sine waves
sum of cosines
Johnson noise, and cosine plus Johnson noise
bandwidth-limited noise
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2. The Instrument and Math
The instrument for radio frequency auto-correlation is shown at the right. All components are passive, i.e. there are no amplifiers.
For signals < ~10 kHz, the input is converted into digital form. The multiplication and integration are then performed numerically.
The autocorrelation of a signal is given by the following integral.
The output of the averager as a function of delay is called an autocorrelogram. Note the similarity to convolution - only the t+t term differs!
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3. Properties
The device only works well when T can become very large. For radio frequency signals this requires continuous, very slowly changing signals. For signals that are digitized, this requires very large data sets.
In many applications the output is measured with a fixed value of t that minimizes noise and maximizes the signal.
Auto-correlation can remove huge amounts of noise and is particularly useful for frequencies too high for any common laboratory measuring device.
For repetitive signals, autocorrelation can be used to shift very high frequencies to arbitrarily low frequencies. This is done by controlling the rate at which t is scanned.
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4. Autocorrelation & Fourier Transforms
The autocorrelogram can be obtained using the Fourier transform basis set. The following is the autocorrelation theorem.
Given
and
then
This makes the determination of C1,1(t) rather easy:
start with F(t) and forward transform to obtain the spectrum
square the spectrum
inverse transform the squared spectrum to obtain the autocorrelogram
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5. Autocorrelation & Fourier Transforms
Proof:
Q.E.D.
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6. Cosine
Given
squaring the spectrum (remember the 1/2 within the definition of dd)
then
Thus, the auto-correlation of a cosine is the same cosine with a modified amplitude.
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7. Sine Given squaring the spectrum then
Thus, the auto-correlation of a sine is a cosine with the same period and a modified amplitude. All autocorrelograms are even functions, distorting the shape of non-even F(t).
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8. Sum of Cosines Given squaring the spectrum then
Or, the autocorrelogram of a sum of cosines is a sum of cosines. Note that the squaring of amplitudes may change the shape of the signal!
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9. Johnson Noise Given Johnson noise
where g is a random phase factor, the squared magnitude of the spectrum becomes
then
Thus, all Johnson noise is packed into an impulse function appearing at zero delay. This makes it extremely easy to remove noise from a sinusoidal frequency.
This noise only goes away when T = ∞.
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10. Cosine with Johnson Noise
For a single measurement, the autocorrelogram can be curve fit to a cosine by dropping the first couple points.
For a continuous measurement, where the primary interest is the amplitude of the cosine as a function of time, the delay is fixed at one of the cosine peaks. The measurement is then made as a function of time using this fixed delay. With real noise a large delay might be necessary, i.e. use a cosine peak far from t = 0.
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11. Bandwidth-Limited Noise
Given truncated noise
the square of the spectrum becomes
then
Given noise limited by a RC low pass filter
the square of the spectrum becomes
then
Because exponential and sinc functions decay slowly, bandwidth limited noise can only be removed with large delays.
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12. Autocorrelation Transfer Function
For a temporal signal, F(t), having a spectrum, F(f ), the delay transfer function is given by an integral involving only F(t) and the delay.
The spectral transfer function is given by the square of F(f ).
As a result the transfer function depends solely upon the signal and not the instrument. The delay, t, needs to be adjusted over a range sufficiently large to cover at least the range of time yielding finite values for F(t). The integration time, T, only affects the degree to which the autocorrelation approaches ideality.
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