Interpretation of Describing Function as an Optimal Quasilinearization
Using a Functional Analytic Approach
Let CF be the space of almost everywhere continuous functions that have finite average power, i.e.,
 T
2
lim  1  d f ()    f  CF and T  (0, ). [It is implied that the limit exists and is finite.]

T  T 0

Let    : CF  CF  be the space of causal time-invariant bounded operators. Note that the operators
 : CF  CF could be linear or nonlinear.
Remark 1: CF is a vector space over the real field  with an inner product defined as:
T
f , g T  1  d g() f () f , g  C F [0, T] and any finite T  (0, ).
T
0
Accordingly, the norm on this inner product space is defined as:
T
f T  1  d f ()
T
2
for any finite T  (0, ) and f  CF .
0
It follows from Cauchy-Schwarz inequality that f , g T  f T g T
f , g  C F .

Definition 1: The cross-correlation of two functions f , g  CF[ is defined as:
 T


 f , g ()  lim  1  d g(t  ) f ( t )


T
T  0

Let L be a subspace of  containing only the linear finite-dimensional operators. That is, L consists of proper
Hurwitz rational transfer functions. For example, let e(t )  CF be the input to the nonlinear function    so that
its output is e( t )  C F . If h  L is a linear approximation of    , then the output corresponding to the input
t
e(t )  CF is h  e ( t )   d h ( t  ) e() and h  L1[0, ) . The problem is to identify a proper Hurwitz transfer
0
function ĥ corresponding to the impulse response h  L that approximates the nonlinear function    in the
sense of minimizing the following error functional:
 T
2
J(h)  lim  1  dt (e(t ))  (h  e)( t ) 


T
T 0

The resulting (causal Hurwitz) impulse response h(t) or the corresponding transfer function ĥ (s) ) is called an
optimal quasi-linearization of the nonlinear operator   .
Theorem 1: The error functional J(h) is minimized if and only if e,he ()  e,(e) ()   0.
 


t
~
~
~
~
Proof: Let h  L be another linear approximation of    . Then, h  e ( t )   d h ( t  ) e() and h  L1[0, ).
0
~ ~
The objective is to identify and optimal h  L such that J(h)  J(h) h  L . Let us use the variational approach to
identify an optimal h .
 T 
2
~
~
~
2 
Let J( h , h )  J( h )  J(h)  lim  1  dt  (e( t ))  ( h  e)( t )  (e( t ))  (h  e)( t )  

T
T 0 
 
 T  ~
2
~

2
 lim  1  dt  ( h  e)( t )  (h  e)( t )  2 ( h  h )  e (t ) (e( t ))  
T  T 0 
 

1





 T  ~
2
~

 lim  1  dt  (h  h )  e ( t )  2 (h  h )  e (t ) (h  e)( t )  (e( t )  
T  T 0 
 
~
~
Since h minimizes the error functional J() if and only if J(h, h)  0 h  L , we must have



 T

~
lim  1  dt ( h  h )  e (t ) (h  e)( t )  (e(t )   0 .

T T 0

Expressing the convolution terms as integrals, we have:
T
1 dt (~
h  h )  e ( t ) (h  e)( t )  (e( t ) 

T



0
T t


~
 1  dt    d ( h  h )( ) e( t  )  (h  e)( t )   (e( t ) 

T


0  0


T
T

~
 1  d (h  h )( )   dt e( t  ) (h  e)( t )  (e( t ) by interchanging the order of integration.


T
0


T
 T ~



~
Therefore, lim  1  d ( h  h )( )   dt e( t  ) (h  e)( t )  (e( t )    0 for all functions h  h  L . This is


T


T  0


T


possible if lim  1  dt e( t  ) (h  e)( t )  (e( t )   0   0 implying that


T
T 

 1 T

lim 
dt (h  e)( t  )  (e(t  )e(t )  0   0 .


T  T 0

 T

 T

Setting T in place of T   when T   , we have lim  1  dt (h  e)( t  ) e( t )   lim  1  dt (e( t  ) e( t ) 
 T T

T T 0

 0

Hence, e,he ()  e,(e) ()   0.



Remark 2: The optimal approximation of the nonlinear function  by a linear transfer matrix ĥ is dependent on
the given reference point. However, there is no guarantee of optimality or even existence of such a transfer function
if the reference point is altered because the nonlinear function  is also altered.

Example 1: Let the input signal be a non-zero constant, i.e., e(t)  c  0 t  0. Then, (e(t))  (c) t  0 and
e,(e) ()  c (c) which is a constant   0. By the optimality condition, the linear approximation h(t) must
satisfy e,he ()  c (c) that leads to:
 T

 T

lim  1  dt (h  e)( t  ) c  c (c)  lim  1  dt (h  e)( t )   (c)


T T 0
T T 0


If the nonlinearity is memoryless, then (c) is a real constant. Then a possible choice is (h  e)( t )  (c) t  0.
This implies that h ( t ) 
 (c)
( t ) or ĥ(s)  (c) . However, this choice may not be unique.
c
c


Example 2: Let e(t)  E Sin( t) t  0. Then, (e( t ))  A 0   A k Sin (k  t )  B k Cos (k  t )  t  0. Let  be
k 1
an odd function implying that A 0  0 and Bk  0 k  N. Then,
 T  

 e, (e) ()  lim  1  dt   A k Sin (k  (t  ) E Sin (k  t )  


T
T  0  k 1

2
Since
2 / 
 A1 T


 dt Sin (k  t ) Sin (  t )    k , we have  e, (e) ()  lim  T  dt Sin ( ( t  ) E Sin (t )  .
T 
0
0

t A
A
A
Now, if we choose ĥ(s)  1  h(t )  1 (t ), then (h  e)( t )   d 1 ( t  ) E Sin ( t )  A1 Sin ( t ) . Then,
E
E
E
0
 T

 T

 e, h e ()  lim  1  dt Sin ( (t  ) E Sin (t )  lim  1  dt Sin ( (t  ) E Sin ( t )   0.
 T  T

T  T 0

 0

Therefore, e,(e )()  e, he ()   0.

Theorem 2: Let the input and output of the nonlinear block be e(t )  E Sin (t ) and (e(t ))  z( t )  z ss (t )  z tr ( t ),
respectively, where the first harmonic z1 is continuous and (2 / )  periodic and z tr  L2 ([0, )) . Let the
nonlinear operator () be approximated by the following transfer function ĥ (s) such that


Re ĥ (i) 
g re()
E


and Im ĥ (i) 
g im ()
E
and gim () are even and odd functions of  , respectively; and the first harmonic of zss (t ) is
where g re()
given as: z1ss ( t )  g re () Sin ( t )  g im () Cos ( t ) .
T
2
Proof: We have z tr  L 2 ([0, )), i.e.,  dt z tr ( t )    ; and the energy of the input signal over any finite T is:
0
T
2
T
2
2
2
 dt e( t )  E  dt Sin ( t )  E T  e(t ) T  E T for any finite T. .
0
T
e, z tr T  1  dt z tr ( t ) e( t )  , it
T
Having
0
0
follows from Cauchy-Schwarz inequality that e, z tr
 e T z tr T for any finite T.
T
E
Therefore,
e, z tr T  lim e T z tr T  lim
 0.
T 
T 
T  T
e, z tr ()  0   0. The implication is that any transient part of the nonlinear block output is uncorrelated to its
Now,
the
cross-correlation
 e, z tr  lim
input.
We
consider
only the
periodic
e, z ()  e, zss ()  e, z tr ()  e, zss () .
Since
e(t)  E Sin ( t), setting
T  2 / 
part
where
zss (t )
of
  N , we have
the
output
z(t)
because
2k ( t  ) 
e( t ), Sin 
  0 and
T

 T
2k ( t  ) 
e( t ), Cos 
for all k  N - 0 . Letting    , it follows that  e, z ss ()   1 () .

T
e, z ss

 T
Given ê(i)  iE(  )  (  ) , the output of the linear approximation block is obtained as the Fourier
inverse of ĥ (i) ê(i) as follows:


(h * e)(t ) = Re( ĥ(i)) Sin ( t )  Im( ĥ(i)) Cos ( t ) E
Therefore, (h  e)( t )  z1ss ( t ) t  0. Proof is thus complete.

Definition 2: Following the notations in Theorem 2, the describing function Keq (,) of the nonlinear operator 
g ( )
g ( )
 i im
is the complex-valued function defined as: K eq (E, )  re
.
E
E

Theorem 3: If the nonlinear operator  is memoryless, time-invariant, and continuous, then the describing function
Keq (E, ) is independent of  .
The output in response to the input e(t)  E Sin(t) is z( t )  E Sin (t ) that is decomposed as
z(t )  zss (t )  z tr (t ) as the steady-state and transient components. Given the operator  to be memoryless, time-
Proof:
3
invariant, almost everywhere continuous, and bounded, the transient part z tr ( t ) of the output becomes zero.
g ( )
g ( )
Therefore, the conditions of Theorem 2 prevail and we have K eq (E, )  re
. Now consider another
 i im
E
E
~
~
signal ~
e (t )  E Sin (t ) of same amplitude and different frequency    so that ~e can be obtained from e by time
~
scaling. Therefore, e( t )  ~e  t  implying that the first harmonics are identical after time scaling. Therefore,

~
K eq (E, )  K eq (E, ).

Corollary 1 to Theorem 3: If, in addition, the operator  is odd, then K eq is real for E>0, i.e., g im ()  0 .
Proof: For e(t )  E Sin (t ) and  being odd, Fourier expansion of e(t ) does not contain any cosine term.

Corollary 2 to Theorem 3: Let, in addition to the conditions specified in Theorem 3 and Corollary 1,  be sectorbounded, i.e.,  k1, k 2  (0, ) such that k1  2   ()  k 2  2   . Then, k1  K eq (E)  k 2 .
Proof: By Corollary 1 to Theorem 3, K eq is real and independent of  because  is memoryless, time-invariant,
(almost everywhere) continuous, and odd. Therefore,
K eq (E)  
E
2 / 
 dt  (E Sin ( t )) Sin ( t ) 
0
2
2
 1  d  (E Sin ) Sin   1  d
E
E 2 0
0
2
Similarly, K eq (E)  1  d (E Sin ) Sin  
E
0


.
k1 E Sin 2  k1


2
2
1
 d k 2 E Sin   k 2
E 2 0
4
