AN0001
15
AN0001
Optimization of Quadrature Modulator Performance
Optimization of Quadrature Modulator Performance
Introduction
Quadrature modulators are common building blocks in a communications link. By their nature they are capable of send-
ing virtually any modulation scheme. They can send both analog, such as AM and FM, and digital modulation schemes,
such as BPSK, QAM, and QPSK.
A common set of specifications for quadrature modulators is the carrier suppression and sideband suppression. The
question often arises as to how they correlate with amplitude and phase error for the same device. This article presents
a derivation showing the relationship between these four parameters.
Because the overall performance of the system can be affected by the modulator, its performance is important. There-
fore, the need to optimize the carrier suppression and the sideband suppression of a quadrature modulator often arises.
For those who may not be experienced with quadrature modulators, this can be a confusing task. This article will make
this optimization easier to understand and perform.
Discussion
The first question that arises is why optimization is required at all. The primary reason is that there are imbalances in the
Gilbert cell mixers and phase error introduced by the phase-shifting network. These imperfections are caused by slight
differences in devices on the same die. There are also imbalances and offsets between the in-phase and quadrature sig-
nal paths as a result of process variations. These errors are not present in an ideal device, however they cannot be elim-
inated in practice.
The second question then would be how to compensate the device to optimize the performance. To understand how
compensation can be achieved, it is helpful to understand how sideband suppression, carrier suppression, amplitude
error and phase error are related.
The block diagram of a typical quadrature modulator is shown in Figure 1.
In-Phase Signal
Σ
Quadrature Signal
RF Output Signal
LO Signal
0°
90°
15
TECHNICAL NOTES
AND ARTICLES
Figure 1. Quadrature Modulator Functional Block Diagram
The circuit multiplies the in-phase signal by the local oscillator and the quadrature signal by a 90° shifted version of the
local oscillator. These signals are then summed to form the RF output signal.
Copyright 1997-2002 RF Micro Devices, Inc.
15-9
AN0001
Derivation
To begin the derivation, the input signals are defined as follows:
I
(
t
)
=
G
cos
( ωt
+
φ )
+
D
Q
(
t
)
= cos
( ωt
+ 90°
)
Note that this implies that the error injected into the system is due to the in-phase signal. The amplitude G represents the
ratio of the amplitudes of the input signals. The phase offset,
φ,
represents the phase error. This is typically a small value
representing how far from quadrature the signals are. The DC offset, D, represents the DC offset between the two sig-
nals. Ideally there is no offset, but in practice, there is an offset introduced by imbalances in the modulator circuitry. The
quadrature signal is, by definition, 90° out of phase with the in-phase signal.
With the signals defined, it is a matter of performing the mathematical analysis that parallels the modulators’ functional-
ity. The RF output is given by
RF
(
t
)
=
G
cos
( ωt
+
φ )
cos
( ω
c
t
)
+
D
cos
( ω
c
t
)
– sin
( ω
c
t
)
cos
( ωt
+ 90°
)
As can be seen, the carrier can only be present at the output if there is a DC offset between the input signals. Now, rear-
range the terms to get the upper sideband and lower sideband terms using the trigonometric identities
1
1
1
-
-
sin
( α )
cos
( β )
= -- sin
( α
–
β )
+ -- sin
( α
+
β )
2
2
1
1
-
-
cos
( α )
cos
( β )
= -- cos
( α
–
β )
+ -- cos
( α
+
β )
2
2
1
1
-
-
USB
(
t
)
= --
G
cos
( ω
c
t
+
ωt
+
φ )
– -- sin
( ω
c
t
+
ωt
+ 90°
)
2
2
1
1
-
-
LSB
(
t
)
= --
G
cos
(
–
ω
c
t
+
ωt
+
φ )
– -- sin
( ω
c
t
–
ωt
– 90°
)
2
2
These can be rewritten to obtain
1
1
-
-
USB
(
t
)
= --
G
cos
( ω
c
t
+
ωt
+
φ )
– -- cos
( ω
c
t
+
ωt )
2
2
1
1
-
-
LSB
(
t
)
= --
G
cos
( ωt
–
ω
c
t
+
φ )
+ -- cos
( ω
c
t
–
ωt )
2
2
Applying the trigonometric identity
1
15
TECHNICAL NOTES
AND ARTICLES
cos
( α
+
β )
= cos
( α )
cos
( β )
– sin
( α )
sin
( β )
with
α
=
ω
c
t
+
ωt
β
=
φ
the final form of the equations is reached
1
1
1
-
-
-
USB
(
t
)
= --
G
cos
( ω
c
t
+
ωt )
cos
( φ )
– --
G
sin
( ω
c
t
+
ωt )
sin
( φ )
– -- cos
( ω
c
t
+
ωt )
2
2
2
1
1
1
-
-
-
LSB
(
t
)
= --
G
cos
( ωt
–
ω
c
t
)
cos
( φ )
– --
G
sin
( ωt
–
ω
c
t
)
sin
( φ )
+ -- cos
( ω
c
t
–
ωt )
2
2
2
15-10
Copyright 1997-2002 RF Micro Devices, Inc.
AN0001
We want these equations in this form is to allow them to be easily converted to envelope-phase form
2
since the sideband
suppression is the ratio of the magnitude (envelope) of the upper sideband to the magnitude of the lower sideband. In
general, a signal can be represented as the sum of the real and imaginary terms
x
(
t
)
=
x
R
(
t
)
cos
( ωt )
–
x
I
(
t
)
sin
( ωt )
This expression can be rewritten in envelope-phase form as follows
x
(
t
)
=
r
(
t
)
cos
[ ωt
+
φ (
t
) ]
where
r
(
t
)
=
x
R
(
t
)
–
x
I
(
t
)
2
2
x
I
(
t
)
-----------
-
φ (
t
)
= atan
x
R
(
t
)
The upper sideband and lower sideband envelopes are
USB
env
=
LSB
env
=
After expanding these terms we obtain
2
2
1
G
cos
φ
– 1
+
– 1
G
sin
φ
--
-
--
-
--
-
2
2
2
2
2
1
G
cos
φ
+ 1
+
– 1
G
sin
φ
--
-
--
-
--
-
2
2
2
USB
env
=
LSB
env
=
The ratio for sideband suppression is then given by
1
2
1
1
--
G
– --
G
cos
φ
+ --
-
-
-
4
2
4
1
2
1
--
G
+ --
G
cos
φ
+ 1
-
-
--
-
4
2
4
USB
env
------------------ =
-
LSB
env
1
2
1
1
--
G
– --
G
cos
φ
+ --
-
-
-
4
2
4
---------------------------------------------- =
1
2
1
-
--
-
--
G
+ --
G
cos
φ
+ 1
-
2
4
4
G
– 2G cos
φ
+ 1
------------------------------------------
2
G
+ 2G cos
φ
+ 1
2
Therefore, the lower sideband suppression in dBc (decibels relative to the upper sideband) is given by
15
TECHNICAL NOTES
AND ARTICLES
G
– 2G cos
φ
+ 1
G
– 2G cos
φ
+ 1
Suppression
(
dBc
)
= 20 log ------------------------------------------ = 10 log ------------------------------------------
2
2
G
+ 2G cos
φ
+ 1
G
+ 2G cos
φ
+ 1
This expression can be plotted as a set of suppression contours,
3
with amplitude error and phase error as the axes. To
facilitate this, the equation was solved for phase error,
φ,
in terms of amplitude error, G, and sideband suppression, SBS.
2
2
–
G
10
+
G
1 – 10
φ
= acos ----------------------------------------------------------------
SBS
2G10
---------
-
10
SBS
---------
-
10
2
SBS
---------
-
10
2
+ 2G
Copyright 1997-2002 RF Micro Devices, Inc.
15-11
AN0001
This equation was used in conjunction with the sideband suppression equation to generate data using a spreadsheet
program. This data was plotted and the result is contained in Figure 2.
12
-20dBc
10
-22dBc
8
Phase Error (degrees)
-24dBc
6
-26dBc
4
-28dBc
-30dBc
2
-32dBc
-34dBc
-38dBc
-36dBc
-40dBc
0
0.0
0.1
0.2
0.3
0.4
0.5
Amplitude Error (dB)
0.6
0.7
0.8
0.9
1.0
Figure 2. Sideband Suppression vs. Amplitude Error and Phase Error
Since it is often easiest to balance the amplitude error, it is convenient to plot phase error versus sideband suppression
assuming the amplitude error has been balanced. Figure 3 is such a plot.
15
TECHNICAL NOTES
AND ARTICLES
15-12
Copyright 1997-2002 RF Micro Devices, Inc.
AN0001
12
10
8
Phase Error (degrees)
6
4
2
0
-20
-25
-30
-35
Sideband Suppression (dB)
-40
-45
-50
Figure 3. Sideband Suppression vs. Phase Error with no Amplitude Error
Optimization Example
The RF2422, an HBT 2.5GHz direct quadrature modulator, and its fully assembled evaluation board were chosen to
demonstrate one way of optimizing performance. The board allows quick testing of the device, requiring two DC connec-
tions, a local oscillator input, two baseband inputs (In-Phase and Quadrature), and the RF output. All but the DC inputs
are standard SMA style connectors. Refer to the schematic in Figure 4.
C2
SMA
100 pF 50
Ω
Microstrip
J4
P1-1
C2
100 p
F
C1
0.33 pF
1
2
3
4
P1-3
SMA
J2 50
Ω
Microstrip
C5
33 pF
5
6
7
POWER
CONTROL
14
13
12
11
10
+45°
-45°
SMA
J1 50
Ω
Microstrip
15
C4
100 pF
SMA
J3
50
Ω
Microstrip
R1
56
Ω
9
8
Figure 4. RF2422 Evaluation Board Schematic
Copyright 1997-2002 RF Micro Devices, Inc.
15-13
TECHNICAL NOTES
AND ARTICLES
Σ
京公网安备 11010802033920号
Copyright © 2005-2026 EEWORLD.com.cn, Inc. All rights reserved
电子工程世界
