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
8.6 : 1/11
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.
af(t)
f(t)
power
divider
af(t)
af(t+τ)
mixer
1
/ 9 f(t)·f(t+τ)
1
time
delay
/ 9 C1,1(τ)
averager
The autocorrelation of a signal is given by the following integral.
⎡1T
⎤
C1,1 (τ ) = lim ⎢ ∫ F1 ( t ) F1 (τ + t ) dt ⎥
T →∞ ⎢ T
⎥⎦
⎣ 0
The output of the averager as a function of delay is called an
autocorrelogram. Note the similarity to convolution - only the τ+t
term differs!
8.6 : 2/11
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 τ
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 τ is scanned.
8.6 : 3/11
Autocorrelation & Fourier Transforms
The autocorrelogram can be obtained using Fourier transform
pictures. The following is the autocorrelation theorem.
Given
and
then
F (t ) ↔ Φ ( f )
⎡1T
⎤
C1,1 (τ ) = lim ⎢ ∫ F ( t ) F (τ + t ) dt ⎥
T →∞ ⎢ T
⎥⎦
⎣ 0
C1,1 (τ ) ↔ Φ ( f )
2
This makes the determination of C1,1(τ) 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
8.6 : 4/11
Cosine
(
Given
)
( )
A cos 2π t t 0 ↔ Aδδ + f 0
squaring the spectrum (remember the 1/2 within the definition of δδ)
Aδδ
then
+
(f )
0
(
2
( )
A2 + 0
=
δδ f
2
)
( )
A2
A2 + 0
0
C1,1 (τ ) =
cos 2π τ t ↔
δδ f
2
2
Thus, the auto-correlation of a cosine is the same cosine with a
modified amplitude.
Autocorrelogram
Temporal Waveform
20
5
amplitude
10
C11
0
0
10
5
8.6 : 5/11
0
5
10
time
20
0
0.5
1
τ
1.5
2
Sine
(
Given
)
( )
A sin 2π t t 0 ↔ Aδδ − f 0
squaring the spectrum
Aδδ
then
−
(f )
0
(
2
( )
A2 + 0
=
δδ f
2
)
( )
A2
A2 + 0
0
C1,1 (τ ) =
cos 2π τ t ↔
δδ f
2
2
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).
Autocorrelogram
Temporal Waveform
5
4
amplitude
2
C11
0
0
2
4
8.6 : 6/11
0
5
10
time
5
0
0.5
1
τ
1.5
2
Sum of Cosines
(
)
(
)
( )
( )
A1 cos 2π t t10 + A2 cos 2π t t20 ↔ A1δδ + f10 + A2δδ + f 20
Given
squaring the spectrum
A1δδ
then
+
( ) + A2δδ ( )
+
f10
(
)
f 20
2
( )
( )
A12 + 0
A22 + 0
=
δδ f1 + δδ f 2
2
2
(
)
( )
( )
A12
A22
A12 + 0
A22 + 0
0
0
C1,1 (τ ) =
cos 2π τ t1 +
cos 2π τ t2 ↔
δδ f1 + δδ f 2
2
2
2
2
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!
Autocorrelogram
Temporal Waveform
10
5
amplitude
5
C11
0
0
5
5
8.6 : 7/11
0
5
10
time
10
0
0.5
1
τ
1.5
2
Johnson Noise
m
Φ( f ) = A
Given Johnson noise
the square of the spectrum becomes
Φ ( f ) = A2
2
then
C1,1 (τ ) = A2δ ( 0 ) ↔ A2
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.
Autocorrelogram
Temporal Waveform
1
This noise only
goes away
when T = ∞.
4
amplitude
2
0.5
C11
0
0
2
4
8.6 : 8/11
0
5
10
time
0.5
0
0.5
1
τ
1.5
2
Cosine with Johnson Noise
m
Autocorrelogram
Temporal Waveform
2
4
2
amplitude
1
0
C11
2
0
4
6
0
5
10
time
1
0
0.5
1
1.5
2
τ
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 τ = 0.
8.6 : 9/11
Bandwidth-Limited Noise
Given truncated noise
( )
Φ ( f ) = A rect f 0
the square of the spectrum becomes
then
C1,1 (τ ) =
A2
t0
(
)
( )
Φ ( f ) = A2 rect f 0
2
( )
sinc π τ t 0 ↔ A2 rect f 0
( )
2
the square of the spectrum becomes
Φ ( f ) = A2 lorenz ( f 0 )
C1,1 (τ ) = A2 exp ( − π t t 0 ) ↔ A2 lorenz ( f 0 )
then
Given noise limited by a RC low pass filter
Φ ( f ) = A lorenz f 0
Because exponential and sinc functions decay slowly, bandwidth
limited noise can only be removed with large delays.
8.6 : 10/11
Autocorrelation Transfer Function
For a temporal signal, F(t), having a spectrum, Φ(f ), the delay
transfer function is given by an integral involving only F(t) and the
delay.
⎡1T
⎤
C1,1 (τ ) = lim ⎢ ∫ F ( t ) F (τ + t ) dt ⎥
T →∞ ⎢ T
⎥⎦
⎣ 0
The spectral transfer function is given by the square of Φ(f ).
C1,1 (τ ) ↔ Φ ( f )
2
As a result the transfer function depends solely upon the signal
and not the instrument. The delay, τ, 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.
8.6 : 11/11