EE603 Class Notes
09/16/14
John Stensby
Appendix 9C: Low-Pass Equivalent and Analytic Signal
We start with a wide-sense stationary (WSS), narrow-band Gaussian process
(t)  c (t) cos c t  s (t) sin c t .
(9C-1)
Note that both c and s are zero-mean, WSS low-pass Gaussian processes, as shown in Chapter
9 of the class notes. In what follows, we define the low-pass equivalent and analytic signal
corresponding to (t). Finally, we use this information to select the optimum value of c for a
Gaussian narrow-band noise process.
Low-Pass Equivalent/Complex Envelope
The low-pass equivalent of (9C-1) is defined as
LP (t)  c (t)  js (t) .
(9C-2)
Often, this is referred to as the complex-envelope representation of . Note that LP is a WSS
low-pass Gaussian process. The original band-pass process (t) is related to LP by
(t)  Re LP (t)e jc t  .


(9C-3)
In the analysis of band-pass signals and systems, very often LP is easier to work with than 
since manipulation of messy trigonometric functions/identities is not required (especially true
when computing the band-pass output of a band-pass system).
Analytic Signal
The analytic signal for (t) is defined as
P (t)  (t)  jˆ (t) ,
(9C-4)
9C-1
EE603 Class Notes
09/16/14
John Stensby
where ˆ (t) denotes the Hilbert transform of (t). Note that (9C-4) can be written as
1 

P (t)  (t)  2  12 (t)  j
,
2t 

(9C-5)
where
1 (t)  j 1
2
2t
 U()
(9C-6)
(U() is a unit step in the frequency domain). Therefore, the Fourier transform of (9C-5) can be
written as
 p ( )  2  ( )U( ) ,
(9C-7)
where  p ( )  F P (t) and  ( )  F (t) . To construct p, Equation (9C-7) tells us that we
should start with  truncate its negative frequency components, and double the amplitude of its
positive frequency components.
We desire to obtain a relationship between p and LP. Note that
P (t)  (t)  jˆ (t)  Re  LP (t)e jc t   jRe  LP (t){ je jc t }




 Re  LP (t)e jc t   jIm  LP (t)e jc t 




(9C-8)
 LP (t)e jc t .
By examining the Fourier transform of (9C-8), one can see that the low-pass equivalent is the
analytic signal translated to the left by c in frequency (i.e., the analytic signal translated down to
base band).
9C-2
EE603 Class Notes
09/16/14
John Stensby
Autocorrelation and Crosscorrelation of Complex-Valued Signals
Chapter 7 of the class notes gave a definition for the autocorrelation function of a realvalued, WSS random process x(t). This definition must be modified slightly to cover the more
general case when x(t) is complex valued. For a complex-valued, WSS process x(t), we define
the autocorrelation as
R x ()  E  x(t  )x* (t)  ,


(9C-9)
where the star denotes complex conjugate.
Note that Rx is conjugate symmetric in that
R x ( )  R x ( ) . Of course, if x(t) is real-valued, then so is Rx , and we have R x ( )  R x ( )
= Rx() Finally, power spectrum Sx() = F [Rx] must be real-valued and nonnegative; it is
even if x(t) is real valued.
In a similar manner, let x(t) and y(t) be complex-valued, jointly wide sense stationary
random processes. The crosscorrelation function is defined here as
R xy ()  E  x(t  )y (t)  .


(9C-10)
In general, (9C-11) does not exhibit conjugate symmetry; however, R xy     R*yx    . Cross
spectrum Sxy() = F [Rxy] can be complex valued with negative real/imaginary components.
Autocorrelation function of LP and P
The autocorrelation function of complex-valued, low-pass equivalent LP is
R LP ( )  E  LP (t  )LP (t)   E {c (t  )  js (t  )}{c (t)  js (t)}


 E c (t  )c (t)  s (t  )s (t)  jE s (t  )c (t)  c (t  )s (t)

(9C-12)

 R c ( )  R s ( )  j R c s ( )  R s c ( ) .
9C-3
EE603 Class Notes
09/16/14
John Stensby
However, from Chapter 9, we know that R c ( )  R s ( ) and R s c ( )  R c s ( )
  R c s () . Hence, we can write (9C-12) as


R LP ( )  2 R c ( )  jR c s ( ) .
(9C-13)
In a similar manner, we can write
R p ( )  E  p (t  )p (t)   E {(t  )  jˆ (t  )}{(t)  jˆ (t)}


 R  ( )  j  R 
ˆ ( )  R ˆ ( )   R 
ˆ ( )

(9C-14)

 2 R  ( )  jRˆ  ( ) .
Finally, we can use (9C-8) and write a relationship between R p () and R LP () as
R p ()  E  p (t  )n p (t)   E LP (t  )e jc (t ) nLP (t)e jc t 




 E  LP (t  )nLP (t)  e jc 


(9C-15)
 R LP ()e jc  .
Power Spectral Densities
Equations (9C-13) and (9C-14) have Fourier transforms given by
SLP ()  2 Sc ()  2 jSc s ()
(9C-16)
Sp ( )  4 S ()U() ,
(9C-17)
9C-4
EE603 Class Notes
09/16/14
John Stensby
respectively.
Note that Sc s ()  F [R c s ( )] is a cross-spectral density; it is purely imaginary and
odd in  (since R c s () is an odd function of ). Therefore, j Sc s () is real valued and odd
in (after all, we know that SLP ( ) must be real valued!). Finally, note that (9C-17) implies
4 S ()  Sp ()  Sp () .
(9C-18)
Equation (9C-15) has a Fourier transform given by
Sp ()  SLP (  c ) ,
(9C-19)
where SLP  F [RLP ] and Sp  F [Rp ] are real-valued, non-negative power spectrums of the
low-pass equivalent and analytic signal, respectively. Equation (9C-19) shows that the power
spectrum of the analytic signal can be obtained by translating up to c the power spectrum of the
low-pass equivalent.
Optimum Value of c for Use in Band-Pass Model
Given a band-pass process (t), representation (9C-1) is not unique. That is, there is a
range of c values that could be used, each value accompanied by a different set of low-pass
functions c(t) and s(t) (i.e., c and s depends on the value of c that is used in the band-pass
model). However, for a given band-pass process (t), it is possible to define and compute an
optimum value of c. This is accomplished in what follows.
Clearly, the magnitude of the low-pass equivalent, LP is the actual envelope of noise
(9C-1). Note that LP is dependent on the value of c that is used in (9C-1). In what follows, the
optimum c is defined as that value which produces the least average temporal variation in the
low-pass equivalent. That is, the optimum value of c minimizes E[dLP/dt2], a quantity that
does not depend on time. Equivalently, the optimum value of c minimizes the RMS value of
dLP/dt.
9C-5
EE603 Class Notes
09/16/14
John Stensby
Now, the power spectrum of dLP/dt is 2 SLP ( )  2 Sp (   c ) , a result that follows
from (9C-19). Hence, the optimum value of c minimizes
 d
E  LP
 dt
2
1  2
1 

 Sp (  c )d 
(  c )2 Sp ()d .


2 
 2 
(9C-20)
With respect to c, differentiate (9C-20), and set the derivative equal to zero. This produces the
constraint
1 
2(  c ) Sp ()d  0 .
2 
(9C-21)
Finally, the optimum value of c is

  Sp () d .
c  

 Sp () d
(9C-22)
Note that (9C-22) is the centroid of Sp () .
Example 9C-1: Consider the noise with spectrum depicted by Fig. 9C-1a). From (9C-19), we
know that Sp has its spectrum concentrated in a narrow band centered at +c, a positive
a)
S()
1

b)

0
Sp ( )
0





4

Fig. 9C-1: a) Power spectrum of narrow band noise. b) Power spectrum
of the corresponding analytic signal.
9C-6
EE603 Class Notes
09/16/14
John Stensby
number. From (9C-18), we can immediately plot Sp as Fig. 9C-1b). From (9C-22), we
calculate the optimum
c 
4
2
d
1
2
4
1
d

1  2
2 2
 12 

2  1

2  1
,
2
(9C-23)
as expected.
9C-7