Complex Multiplier/Mixer
Megafunction
Solution Brief 4 November 1996, ver. 1
A-SB-004-01
Altera Corporation
ALTERA MEGAFUNCTION PARTNERS PROGRAM
ALTERA MEGAFUNCTION PARTNERS PROGRAM
Target Applications:
Communications
Digital Signal Processing
Family
: FLEX 10K and FLEX 8000
Vendor
:
Nova Engineering, Inc.
5 Circle Freeway Drive
Cincinnati, OH 45246
Tel. (513) 860-3456
Fax (513) 860-3535
E-mail sales@nova-eng.com
WWW http://www.nova-eng.com
Features
■
Parallel implementation for maximum speed
■
Parameterized data width
■
Optimized for the Altera
®
FLEX
®
10K and FLEX 8000 device architectures
■
Ideal for the following applications:
– Digital mixers
– Discrete Fourier transforms
– Modulators/demodulators
– Complex arithmetic
– Wireless telecommunication systems
General Description
The complex multiplier/mixer megafunction from Nova Engineering can multiply two
complex numbers or mix two complex signals. This function can be used for vector
cross products, vector dot products, up/down frequency conversion, differential phase
detection, digital amplitude modulation (AM), and quadrature amplitude modulation
(QAM). Nova Engineering will customize the input width, output width, and
processing latency of the complex multiplier/mixer megafunction at no additional
cost. See Figure 1.
Figure 1. Complex Multiplier/Mixer Megafunction Block Diagram
Functional Description
The complex multiplier/mixer megafunction is a parallel and pipelined architecture
that provides maximum speed. The function’s internal and external operations are
synchronized to the rising clock edge, and an asynchronous
reset
input is provided
for initializing all internal registers.
+
–
+
+
a[
n
..0]
b[
n
..0]
c[
n
..0]
d[
n
..0]
clk
reset
real[
m
..0]
imag[
m
..0]
a[
n
..0]
d[
n
..0]
b[
n
..0]
c[
n
..0]
Complex Multiplier/Mixer
Megafunction
SB 4: Complex Multiplier/Mixer Megafunction
2 Altera Corporation
The complex multiplier/mixer megafunction computes the following function:
real[
m
..0]
+
j
imag[
m
..0]
=
(
a[
n
..0]
+
j
b[
n
..0]
)
×
(
c[
n
..0]
+
j
d[
n
..0]
)
where:
real[
m
..0]
= (
a[
n
..0]
×
c[
n
..0]
) – (
b[
n
..0]
×
d[
n
..0]
)
imag[
m
..0]
= (
a[
n
..0]
×
d[
n
..0]
) + (
b[
n
..0]
×
c[
n
..0]
)
Because
b[
n
..0]
and
d[
n
..0]
are both imaginary (i.e., preceded by
j
= ), the
real[
m
..0]
output is equal to (
a
[
n
..0]
×
c
[
n
..0])
– (
b
[
n
..0]
×
d
[
n
..0]
)
. The
imag[
m
..0]
output is implicitly imaginary (i.e., preceded by
j
).
Ports
This function has input ports for two complex numbers or data streams. Table 1
describes the ports for the complex multiplier/mixer megafunction.
Parameters
Nova Engineering can customize the width of the input and output data parameters to
eliminate unnecessary logic and precision, and to optimize speed and area. In addition,
the multiplier latency parameter can be increased to improve execution speed or
reduced to conserve logic cells. See Table 2. The total latency from input to output is
equal to the specified multiplier latency plus one. The output of the adders is latched
to increase the speed without increasing the logic used.
Performance
The megafunction does not require embedded array blocks (EABs), and thus performs
efficiently in both FLEX 10K and FLEX 8000 device families. Because this function does
not use EABs, it can be combined with a sine/cosine look-up table (LUT) for frequency
down conversion or discrete Fourier transform (DFT) functions within a single device.
1–()
Table 1. Complex Multiplier/Mixer Megafunction Ports
Name Type Size Description
a[
n
..0]
Input 4 to 12 bits Real part of complex vector 1. Two’s complement format.
b[
n
..0]
Input 4 to 12 bits Imaginary part of complex vector 1. Two’s complement format.
c[
n
..0]
Input 4 to 12 bits Real part of complex vector 2. Two’s complement format.
d[
n
..0]
Input 4 to 12 bits Imaginary part of complex vector 2. Two’s complement format.
clk Input – System clock.
reset Input – Asynchronous reset, active high.
real[m..0] Output 4 to 25 bits Real part of complex result. Two’s complement format.
imag[m..0] Output 4 to 25 bits Imaginary part of complex result. Two’s complement format.
Table 2. Complex Multiplier/Mixer Megafunction Parameters
Parameter Typical Value Description
Input data width 4 to 12 bits Specifies the width of inputs a[n..0], b[n..0], c[n..0], and d[n..0].
Output data width 4 to 25 bits Specifies the width of the output data. Truncation retains the most significant bits
(MSBs) of the result and discards the least significant bits (LSBs).
Multiplier latency 0 to 4 clocks Specifies the number of pipeline registers added to the design. Pipelining increases
the operating speed at the expense of additional logic cells.
SB 4: Complex Multiplier/Mixer Megafunction
Altera Corporation 3
Typically, the complex multiplier/mixer megafunction is used as a coprocessor to a
digital signal processing (DSP) function. The efficiency of the DSP processor design can
be improved by implementing both the megafunction and a FIFO function into the
same device. For example, the input/output process loading on the DSP processor can
be reduced if the megafunction’s inputs and outputs are accessed as blocks of data
through a FIFO function.
Table 3 shows the device utilization and maximum clock frequency for a typical
implementation of the complex multiplier/mixer megafunction.
Contact Nova Engineering for more information on the complex multiplier/mixer
megafunction.
Applications
The complex multiplier/mixer megafunction is useful for a variety of applications,
including vector arithmetic, digital AM and QAM, and frequency down conversion.
Vector Arithmetic
The complex multiplier/mixer megafunction can calculate vector cross products and
vector dot products. For example, to calculate the vector cross product of vectors
R = rx + ry and S = sx + sy, each variable is assigned to an input of the megafunction
(i.e., a[n..0] = rx, c[n..0] = sy, b[n..0] = rx, and d[n..0] = sy). The
megafunction performs the calculation and provides the result on the real[m..0]
output.
A vector dot product can be calculated in a similar manner. To calculate the vector dot
product of vectors R = rx + ry and S = sx + sy, each variable is assigned to an input of
the megafunction (i.e., a[n..0] = rx, c[n..0] = sy, b[n..0] = ry, and
d[n..0] = sx). The complex multiplier/mixer megafunction performs the calculation
and provides the result on the imag[m..0] output.
Digital AM & QAM
Amplitude modulation (AM) and quadrature amplitude modulation (QAM) require
the source data to vary the amplitude of a carrier frequency over time. For example, to
create an AM wave with the complex multiplier/mixer megafunction, assign the
carrier frequency to the c[n..0] input and the information signal to the a[n..0]
input. Set the b[n..0] and d[n..0] inputs to zero. The megafunction produces an
AM wave on the real[m..0] output.
Table 3. Typical Device Utilization
Implementation Clock (Max) Logic Cells EABs
Input data width = 8 bits
Multiplier output width = 16 bits
Output data width = 8 bits
Multiplier latency = 3 clocks
50 MHz 600 0
SB 4: Complex Multiplier/Mixer Megafunction
Copyright 1996 Altera Corporation. Altera, AMPP, FLEX, FLEX 10K, and FLEX 8000 are trademarks and/or service marks of Altera
Corporation in the United States and other countries. Other brands or products are trademarks of their respective holders. The specifications
contained herein are subject to change without notice. Altera assumes no responsibility or liability arising out of the application or use of any
information, product, or service described herein except as expressly agreed to in writing by Altera Corporation. Altera customers are advised
to obtain the latest version of device specifications before relying on any published information and before placing orders for products or
services. All rights reserved.
4 Altera Corporation
®
2610 Orchard Parkway
San Jose, CA 95134-2020
(408) 894-7000
http://www.altera.com
Frequency Down Conversion
Communications systems often transmit and receive data at frequencies significantly
faster than the data rate of the information. The megafunction can recover this
information or convert it to a lower frequency by mixing a modulated signal with a
carrier frequency. For example, to mix a modulated signal with a carrier frequency
generated from a numerically controlled oscillator (NCO), apply the NCO outputs to
the c[n..0] and d[n..0] inputs of the megafunction. Apply the received signal to
the a[n..0] input.
For example, if y(t) = s(t) × cos(wt) is the received signal, then mixing with the carrier
frequency cos(wt) produces:
z(t) = s(t) × cos(wt) × cos(wt)
Where: s(t) = Information signal
Using the trigonometric identity cos(a) × cos(b) = (1/2) × cos(a – b) + (1/2) × cos(a + b)
and substituting cos(a) = cos(b) = cos(wt) produces the following:
z(t) = s(t) × (1/2) × [cos(a – b) + cos(a + b)]
Low-pass filtering can be accomplished via a low-pass, finite impulse response (FIR)
filter, effectively discarding the cos(a + b) element because it is twice the carrier
frequency. Because a = b, the following equation results:
z(t) = (1/2) × s(t)
The carrier frequency is removed to show the information signal.
