Generalized pencil-of-function method

Generalized pencil-of-function method (GPOF), also known as matrix pencil method, is a signal processing technique for estimating a signal or extracting information with complex exponentials. Being similar to Prony and original pencil-of-function methods, it is generally preferred to those for its robustness and computational efficiency.

The method was originally developed by Yingbo Hua and Tapan Sarkar for estimating the behaviour of electromagnetic systems by its transient response, building on Sarkar's past work on the original pencil-of-function method. The method has a plethora of applications in electrical engineering, particularly related to problems in computational electromagnetics, microwave engineering and antenna theory.

Method

Mathematical basis

A transient signal can be represented as:

y

(

t

)

=

x

(

t

)

+

n

(

t

)

≈

∑

i

=

1

M

R

i

e

s

i

t

+

n

(

t

)

;

0

≤

t

≤

T

,

{\displaystyle y(t)=x(t)+n(t)\approx \sum _{i=1}^{M}R_{i}e^{s_{i}t}+n(t);0\leq t\leq T,}

where

y

(

t

)

{\displaystyle y(t)}

is the observed time-domain signal,

n

(

t

)

{\displaystyle n(t)}

is the signal noise,

x

(

t

)

{\displaystyle x(t)}

is the actual signal,

R

i

{\displaystyle R_{i}}

are the residues (

R

i

{\displaystyle R_{i}}

),

s

i

{\displaystyle s_{i}}

are the poles of the system, defined as

s

i

=

−

α

i

+

j

ω

i

{\displaystyle s_{i}=-\alpha _{i}+j\omega _{i}}

,

z

i

=

e

(

−

α

i

+

j

ω

i

)

T

s

{\displaystyle z_{i}=e^{(-\alpha _{i}+j\omega _{i})T_{s}}}

by the identities of Z-transform,

α

i

{\displaystyle \alpha _{i}}

are the damping factors and

ω

i

{\displaystyle \omega _{i}}

are the angular frequencies.

The same sequence, sampled by a period of

T

s

{\displaystyle T_{s}}

, can be written as the following:

y

[

k

T

s

]

=

x

[

k

T

s

]

+

n

[

k

T

s

]

≈

∑

i

=

1

M

R

i

z

i

k

+

n

[

k

T

s

]

;

k

=

0

,

.

.

.

,

N

−

1

;

i

=

1

,

2

,

.

.