Miscellaneous active electrical nonlinear devices – circuits – and – Specific identifiable device – circuit – or system – Unwanted signal suppression
Reexamination Certificate
2001-08-21
2003-04-15
Le, Dinh T. (Department: 2816)
Miscellaneous active electrical nonlinear devices, circuits, and
Specific identifiable device, circuit, or system
Unwanted signal suppression
C327S558000
Reexamination Certificate
active
06549066
ABSTRACT:
FIELD OF THE INVENTION
The present invention relates in general to electronic filter technology and in particular to complex filters implemented in integrated circuits.
BACKGROUND OF THE INVENTION
Complex filters are useful in certain applications in wireless communications. Complex filters offer selective suppression of positive or negative frequency components of a complex or real signal. This feature of complex filters contrasts with the operation of real filters in that real filters have a transfer function that is symmetric around the direct current (DC) position. The ability of complex filters to suppress positive or negative frequency components enables the suppression of image frequencies of a signal. The suppression of image frequencies is a very important consideration in the design and operation of wireless transceivers.
A review of complex signals and complex filters will be useful to understand the present invention. A real signal may have both a positive frequency component and a negative frequency component. For example, a cosine signal cos (&ohgr;t) equals (e
j&ohgr;
+e
−j&ohgr;
)/2 and a sine signal sin (&ohgr;t) equals (e
j&ohgr;
+e
−j&ohgr;
)/2j. The letter j represents the square root of minus one. That is, j
2
=−1. The letter j therefore represents one imaginary unit.
A complex signal is a signal that is composed of two real signals in which one of the real signals is multiplied by j. A complex signal therefore has the form:
x
(
t
)=
x
r
(
t
)+
j x
i
(
t
) (1)
where x
r
(t) is a real signal that represents the real component of the complex signal x(t) and where x
i
(t) is a real signal that represents the imaginary component of the complex signal x(t).
A complex signal x(t) may be amplified by multiplication by a complex constant A+j B. For example, let y(t) be the result of multiplying the complex signal x(t) by the complex constant A+j B. Then
y
(
t
)=(
A+j B
)
x
(
t
) (2)
where
y
(
t
)=
y
r
(
t
)+
j y
i
(
t
). (3)
The expression Y
r
(t) represents a real signal that is the real component of the complex signal y(t) and the expression y
i
(t) represents a real signal that is the imaginary component of the complex signal y(t).
Substituting Equation (1) into Equation (2) and multiplying and equating the real and imaginary parts of the result with y(t) gives:
Y
r
(
t
)=
A x
r
(
t
)−
B x
i
(
t
) (4)
and
y
i
(
t
)=
B x
r
(
t
)
+A x
i
(
t
). (5)
Similarly, a complex signal x(t) may be multiplied by another complex signal z(t) where
z
(
t
)=
Z
r
(
t
)
+j z
i
(
t
). (6)
The multiplication of x(t) by z(t) is represented by:
y
(
t
)=
z
(
t
)·
x
(
t
) (7)
The result of multiplying x(t) by z(t) may be obtained by substituting Equation (6) into Equation (7) and multiplying and equating the real and imaginary parts of the result with y(t). The result is:
y
r
(
t
)=
z
r
(
t
)
x
r
(
t
)−
z
i
(
t
)
x
i
(
t
) (8)
and
y
i
(
t
)=
Z
r
(
t
)
x
r
(
t
)+
z
i
(
t
)
x
i
(
t
). (9)
Complex signals may be filtered by real filters or by complex filters. A real filter has a real impulse response h
r
(t). The transfer function H
r
(j&ohgr;) is a rational polynomial function of j&ohgr;. The transfer function H
r
(j&ohgr;) can be real only if H
r
(j&ohgr;)=H
r
*(−j&ohgr;).
A complex filter has a complex impulse response
h
(
t
)=
h
r
(
t
)+
j h
i
(
t
) (10)
and a complex transfer function
H
(
j&ohgr;
)=
H
r
(
j&ohgr;
)+
j H
i
(
107
). (11)
The response of a linear time invariant system to an arbitrary input x(t) can be expressed as the convolution of x(t) and the impulse response h(t) of the system. That is,
y
(
t
)=
h
(
t
)◯
x
(
t
) (12)
where the symbol ◯ represents the convolution operation. Applying the well known time convolution theorem of the Fourier transform to Equation (12) gives:
Y
(
j&ohgr;
)=
H
(
j&ohgr;
)·
x
(
j&ohgr;
) (13)
where Y(j&ohgr;) is a complex output signal and X(j&ohgr;) is a complex input signal. H(j&ohgr;) is a rational complex polynomial that is a function of j&ohgr;.
Because the input signal X(j&ohgr;) is complex, X(j&ohgr;) is composed of a real part and an imaginary part.
X
(
j&ohgr;
)=
X
r
(
j&ohgr;
)+
j X
i
(
j&ohgr;
). (14)
where X
r
(j&ohgr;) represents the real part of X(j&ohgr;) and where X
i
(j&ohgr;) represents the imaginary part of X(j&ohgr;). Similarly, because the output signal Y(j&ohgr;) is also complex, Y(j&ohgr;) is also composed of a real part and an imaginary part.
Y
(
j&ohgr;
)=
Y
r
(
j&ohgr;
)+
j Y
i
(
j&ohgr;
). (15)
where Y
r
(j&ohgr;) represents the real part of Y(j&ohgr;) and where Y
i
(j&ohgr;) represents the imaginary part of Y(j&ohgr;).
Substituting Equations (11), (14) and (15) into Equation (13) and multiplying and equating the real and imaginary parts of the result to the real and imaginary parts of Y(j&ohgr;) gives:
Y
r
(
j&ohgr;
)=
H
r
(
j&ohgr;
)−
X
r
(
j&ohgr;
)−
H
i
(
j&ohgr;
)
X
i
(
j&ohgr;
) (14a)
Y
i
(
j&ohgr;
)=
H
r
(
j&ohgr;
)
X
r
(
j&ohgr;
)+
H
i
(
j&ohgr;
)
X
i
(
j&ohgr;
) (15a)
In the time domain Equations (14a) and (15a) give:
y
r
(
t
)=
h
r
(
t
)◯
x
r
(
t
)−
h
i
(
t
)◯
x
i
(
t
) (16)
y
i
(
t
)=
h
r
(
t
)◯
x
r
(
t
)+
h
i
(
t
)◯
x
i
(
t
) (17)
where the symbol ◯ represents the convolution operation.
The equation of a transfer function having a complex pole has the form:
H
(
j
ω
)
=
A
s
+
(
p
±
j
q
)
(
18
)
The letter A represents a constant. The letter s represents the quantity j&ohgr;. The letter p represents the real part of the complex pole and the letter q represents the imaginary part of the complex pole. Substitution of Equation (18) into Equation (13) gives:
Y
(
j
ω
)
=
A
s
+
(
p
±
j
q
)
·
X
(
j
ω
)
(
19
)
One of the main applications for complex filters is the selective suppression of positive or negative frequency components of a complex or real signal. This may be accomplished by using a bandpass filter that is obtained from the linear frequency transformation of a lowpass filter. A complex lowpass filter that is centered on the direct current (DC) value (i.e., j&ohgr;=0) of the j&ohgr; axis of a H(j&ohgr;)/j&ohgr; plane may be linearly transformed to create a complex bandpass filter that is centered on another value, j&ohgr;
c
, of the j&ohgr; axis.
Using the linear transformation
s=j&ohgr;−j&ohgr;
c
(20)
will result in a bandpass filter that has the form of the lowpass filter but is centered around the frequency &ohgr;
c
. This form of bandpass filter has only the frequency shifted lowpass filter characteristics for positive frequencies. The transfer function of this form of bandpass filter suppresses negative frequency components.
Substituting Equation (20) into Equation (19) leads to the following design equations for the real and imaginary parts of the output signal Y(j&ohgr;). The argument j&ohgr; in the expressions Y(j&ohgr;) and X(j&ohgr;) in Equations (21) and (22) will be omitted for clarity.
Y
r
=
X
r
-
ω
C
A
Y
i
-
p
A
Y
r
±
q
A
Y
i
j
ω
A
(
21
)
Y
i
=
X
i
-
ω
C
A
Y
r
-
p
A
Y
i
∓
q
A
Y
r
j
ω
A
(
22
)
It would be desirable to provide circuitry on an integrated circuit that is capable of implementing a complex filter of the type represented by a transfer function having a complex pole. In particular, it would be desirable to provide an apparatus for providing the real part Y
r
(j&ohgr;) and imaginary part Y
i
(j&ohgr;) of an output signal Y(j&ohgr;) that results from multiplying an input si
Davis Munck, P.C.
Le Dinh T.
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