MIT OpenCourseWare http://ocw.mit.edu
2.161 Signal Processing: Continuous and Discrete
Fall 2008
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .

Massachusetts Institute of Technology
Department of Mechanical Engineering
2.161
Signal Processing - Continuous and Discrete
Fall Term 2008
Lecture 22 1
Reading:
• Proakis and Manolakis: Secs.
12,1 – 12.2
• Oppenheim, Schafer, and Buck:
• Stearns and Hush: Ch.
13
1 The Correlation Functions (continued)
In Lecture 21 we introduced the auto-correlation and cross-correlation functions as measures of self- and cross-similarity as a function of delay τ .
We continue the discussion here.
1.1
The Autocorrelation Function
There are three basic definitions
(a) For an infinite duration waveform:
φ f f
( τ ) = lim
T →∞
1
T
�
T / 2
− T / 2 f ( t ) f ( t + τ ) d t which may be considered as a “power” based definition.
(b) For an finite duration waveform: If the waveform exists only in the interval t
1 t ≤ t
2 � t
2
ρ f f
( τ ) = f ( t ) f ( t + τ ) d t t
1
≤ which may be considered as a “energy” based definition.
(c) For a periodic waveform: If f ( t ) is periodic with period T
1
�
φ f f
( τ ) =
T t
0 t
0
+ T f ( t ) f ( t + τ ) d t for an arbitrary t
0
, which again may be considered as a “power” based definition.
1 copyright � 2008
22–1

Example 1
Find the autocorrelation function of the square pulse of amplitude a and duration
T as shown below.
The wave form has a finite duration, and the autocorrelation function is
�
T
ρ f f
( τ ) = f ( t ) f ( t + τ ) d t
0
The autocorrelation function is developed graphically below t + r f f
( r f f
( ) = (
- T
�
T − τ
ρ f f
( τ ) =
= a 2
0 a 2 d t
( T τ )
= 0
22–2
− T ≤ τ ≤ T otherwise.

Example 2
Find the autocorrelation function of the sinusoid f ( t ) = sin(Ω t + φ ).
Since f ( t ) is periodic, the autocorrelation function is defined by the average over one period
φ f f
( τ ) =
1
T
� t
0
+ T t
0 f ( t ) f ( t + τ ) d t.
and with t
0
= 0
φ f f
( τ ) =
2
Ω
2
π
�
0
2 π/ Ω
=
1 cos(Ω t ) sin(Ω t + φ ) sin(Ω( t + τ ) + φ ) d t and we see that φ f f
( τ ) is periodic with period 2 π/ Ω and is independent of the phase φ .
1.1.1
Properties of the Auto-correlation Function
(1) The autocorrelation functions φ f f
( τ ) and ρ f f
( τ ) are even functions, that is
φ f f
( − τ ) = φ f f
( τ ) , and ρ f f
( − τ ) = ρ f f
( τ ) .
(2) A maximum value of ρ f f
( τ ) (or φ f f
( τ ) occurs at delay τ = 0,
| ρ f f
( τ ) | ≤ ρ f f
(0) , and | φ f f
( τ ) | ≤ φ f f
(0) and we note that
ρ f f
(0) =
�
∞
−∞ f 2 ( d ) d t is the “energy” of the waveform.
Similarly
φ f f
(0) = lim
T →∞
1
T
�
∞
−∞ f 2 ( t ) d t is the mean “power” of f ( t ).
(3) ρ f f
( τ ) contains no phase information, and is independent of the time origin.
(4) If f ( t ) is periodic with period T , φ f f
( τ ) is also periodic with period T .
(5) If (1) f ( t ) has zero mean ( µ = 0), and (2) f ( t ) is non-periodic, lim ρ f f
τ →∞
( τ ) = 0 .
22–3

1.1.2
The Fourier Transform of the Auto-Correlation Function
Consider the transient case
R f f
(j Ω) =
=
=
=
�
∞
ρ f f
( τ ) e − j Ω τ d τ
�
−∞
∞
��
∞ f ( t ) f ( t + τ ) d t
� e − j Ω τ d τ
� −∞
∞
−∞ f ( t ) e j Ω t d t.
�
F
−∞
( − j Ω)
= | F (j Ω) |
2
F (j Ω)
∞
−∞ f ( ν ) e − j Ω ν d ν or
ρ f f
( τ ) ←→ R f f
(j Ω) = | F (j Ω) | 2 where R f f
(Ω) is known as the energy density spectrum of the transient waveform f ( t ).
Similarly, the Fourier transform of the power-based autocorrelation function, φ f f
�
∞
( τ )
Φ f f
(j Ω) =
=
F
�
{ φ
∞
−∞ f f
( τ ) } =
�
1
�
−∞
T / 2
φ lim
T →∞
T
− T / 2 f f
( τ ) e − j Ω τ d τ f ( t ) f ( t + τ ) d t
� e − j Ω τ d τ is known as the power density spectrum of an infinite duration waveform.
From the properties of the Fourier transform, because the auto-correlation function is a real, even function of τ , the energy/power density spectrum is a real, even function of Ω, and contains no phase information.
1.1.3
Parseval’s Theorem
From the inverse Fourier transform
�
∞
ρ f f
(0) =
∞ f 2 ( t ) d t =
1
2 π or
�
∞
−∞
R f f
(j Ω) dΩ
�
∞ f
∞
2 ( t ) d t =
1
2 π
�
∞
−∞
| F (j Ω) |
2 dΩ , which equates the total waveform energy in the time and frequency domains, and which is known as Parseval’s theorem .
Similarly, for infinite duration waveforms lim
T →∞
�
T / 2 f 2 ( t ) d t =
− T / 2
1
2 π
�
∞
−∞
Φ(j Ω) dΩ equates the signal power in the two domains.
22–4

1.1.4
Note on the relative “widths” of the Autocorrelation and Power/Energy
Spectra
As in the case of Fourier analysis of waveforms, there is a general reciprocal relationship between the width of a signals spectrum and the width of its autocorrelation function.
• A narrow autocorrelation function generally implies a “broad” spectrum
F ( W ) b r o a d a u t o c o r r e l a t i o n n a r r o w s p e c t r u m
• and a “broad” autocorrelation function generally implies a narrow-band waveform.
n a r r o w a u t o c o r r e l a t i o n
F ( W ) b r o a d s p e c t r u m
In the limit, if φ f f
( τ ) = δ ( τ ), then Φ f f
(j Ω) = 1, and the spectrum is defined to be “white”.
F ( W )
" w h i t e " s p e c t r u m i m p u l s e a u t o c o r r e l a t i o n
1.2
The Cross-correlation Function
The cross-correlation function is a measure of self-similarity between two waveforms f ( t ) and g ( t ).
As in the case of the auto-correlation functions we need two definitions:
φ f g
( τ ) = lim
T →∞
1
T
�
T / 2
− T / 2 f ( t ) g ( t + τ ) d τ in the case of infinite duration waveforms, and
�
∞
ρ f g
( τ ) = f ( t ) g ( t + τ ) d τ
−∞ for finite duration waveforms.
22–5

Example 3
Find the cross-correlation function between the following two functions g (
In this case g ( t ) is a delayed version of f ( t ).
The cross-correlation is r f g
T - T where the peak occurs at τ = T
2
− T
1
(the delay between the two signals).
1.2.1
Properties of the Cross-Correlation Function
(1) φ f g
( τ ) = φ gf
( − τ ), and the cross-correlation function is not necessarily an even function.
(2) If φ f g
( τ ) = 0 for all τ , then f ( t ) and g ( t ) are said to be uncorrelated .
(3) If g ( t ) = af ( t − T ), where a is a constant, that is g ( t ) is a scaled and delayed version of f ( t ), then φ f f
( τ ) will have its maximum value at τ = T .
Cross-correlation is often used in optimal estimation of delay, such as in echolocation (radar, sonar), and in GPS receivers.
22–6

Example 4
In an echolocation system, a transmitted waveform s ( t ) is reflected off an object at a distance R and is received a time T = 2 R/c sec.
later.
The received signal r ( t ) = αs ( t − T )+ n ( t ) is attenuated by a factor α and is contaminated by additive noise n ( t ).
t r a n s m i t t e d w a v e f o r m r e c e i v e d w a v e f o r m t - T r e f l e c t i n g o b j e c t v e l o c i t y o f p r o p a g a t i o n : c d e l a y T = 2
�
∞
φ sr
( τ ) =
=
� ∞
∞ s ( t ) r ( t + τ ) d t s ( t )( n ( t + τ ) + αs
∞
= φ sn
( τ ) + αφ ss
( τ − T )
( t − T + τ )) d t and if the transmitted waveform s ( t ) and the noise n ( t ) are uncorrelated, that is
φ sn
( τ ) ≡ 0, then
φ sr
( τ ) = αφ ss
( τ − T ) that is, a scaled and shifted version of the auto-correlation function of the trans­ mitted waveform – which will have its peak value at τ = T , which may be used to form an estimator of the range R .
1.2.2
The Cross-Power/Energy Spectrum
We define the cross-power/energy density spectra as the Fourier transforms of the crosscorrelation functions:
R
Φ f g f g
(j Ω)
(j Ω)
=
=
�
∞
ρ f g
( τ ) e − j Ω τ d τ
� −∞
∞
−∞
φ f g
( τ ) e − j Ω τ d τ.
Then
R f g
(j Ω) =
=
=
�
∞
ρ f g
( τ ) e − j Ω τ d τ
� −∞
∞
� −∞ −∞
∞ f ( t ) e j Ω t d t
−∞
�
∞ f ( t ) g ( t + τ ) e − j Ω τ
�
∞
−∞ g ( ν ) e d t
− j Ω ν d τ d ν
22–7

or
R f g
(j Ω) = F ( − j Ω) G (j Ω)
Note that although R f f
(j Ω) is real and even (because ρ f f
( τ ) is real and even, this is not the case with the in general complex.
cross-power/energy spectra, Φ f g
(j Ω) and R f g
(j Ω), and they are
2 Linear System Input/Output Relationships with Random In­ puts:
Consider a linear system H (j Ω) with a random input f ( t ).
The output will also be random f f
(
( j
Then
Y (j Ω) = F (j Ω) H (j Ω) ,
Y (j Ω) Y ( − j Ω) = F (j Ω) H (j Ω) F ( − j Ω) H ( − j Ω) or
Φ yy
(j Ω) = Φ f f
(j Ω) | H (j Ω) |
2
.
Also
F ( − j Ω) Y (j Ω) = F ( − j Ω) F (j Ω) H (j Ω) , or
Φ f y
(j Ω) = Φ f f
(j Ω) H (j Ω) .
Taking the inverse Fourier transforms
φ yy
( τ ) = φ f f
( τ ) ⊗ F − 1
�
| H (j Ω) |
2
�
φ f y
( τ ) = φ f f
( τ ) ⊗ h ( τ ) .
3 Discrete-Time Correlation
Define the correlation functions in terms of summations, for example for an infinite length sequence
φ f g
( n ) = E { f m g m + n
}
= lim
N →∞
1
�
2 N + 1 m = − N f m g m + n
, and for a finite length sequence
�
ρ f g
( n ) = m = − N f m g m + n
.
22–8

The following properties are analogous to the properties of the continuous correlation func­ tions:
(1) The auto-correlation functions φ
( f f ( n ) and ρ f f
( n ) are real, even functions.
(2) The cross-correlation functions are not necessarily even functions, and
φ f g
( n ) = φ gf
( − n )
(2) φ f f
( n ) has its maximum value at n = 0,
| φ f f
( n ) | ≤ φ f f
(0) for all n.
(3) If { f k
} has no periodic component lim φ n →∞ f f
( n ) = µ 2 f
.
(4) φ f f
(0) is the average power in an infinite sequence, and ρ f f
( n ) is the total energy in a finite sequence.
The discrete power/energy spectra are defined through the z -transform
Φ f f
( z ) = Z { φ f f
( n ) } =
� n = −∞
φ f f
( n ) z − n and
φ f f
( n ) = Z − 1
=
1
2 π j
T
�
�
Φ f f
( z ) }
Φ
π/T f f
( z ) z n − 1 dz
=
2 π
− π/T
Φ f f
( e j Ω T ) e j n Ω T dΩ .
Note on the MATLAB function xcorr() : In MATLAB the function call phi = xcorr(f,g) computes the cross-correlation function, but reverses the definition of the subscript order from that presented here, that is it computes
φ f g
( n ) =
1
M
� f n + m g m
=
− N
1
M
� f n g n − m
− N where M is a normalization constant specified by an optional argument.
Care must therefore be taken in interpreting results computed through xcorr() .
22–9

3.1
Summary of z -Domain Correlation Relationships
(The following table is based on Table 13.2
from Stearns and Hush)
Property
Power spectrum of { f n
}
Cross-power Spectrum
Autocorrelation
Cross-correlation
Waveform power
Linear system properties
Formula
�
Φ f f
( z ) = φ f f
( n ) z − n
Φ f g
( z ) = n = −∞
�
φ f g
( n ) z − n = Φ gf
( z − 1 )
φ
φ
E f f f g
�
Φ yy
Φ f y
(
( n ) = f
( n
2 n z
) =
�
=
) = n = −∞
1
�
2 π j
1
2 π j
φ
H f f
( z
�
)Φ
Φ
Φ f g
(0) =
Y ( z ) = H ( z ) F ( z )
( z ) = H ( z ) H ( z f f f f
− 1
( z ) z
2 π j n − 1
( z ) z n − 1
1
( z )
�
)Φ f f
Φ dz dz
( z ) f g
( z ) z − 1 dz
22–10