Group SD0621

From ECE Department Wiki

Group SD0621 - Frequency Division Multiplexer

This senior design group is in the process of creating a 32 input analog frequency division multiplexer Frequency Division Multiplexer (FDM)/demulitplexer by utilizing Quadrature Amplitude Modulation (QAM). The idea is to take 32 audio input signals modulate them to 16 different frequencies using QAM. The quadrature aspect of the signals allows them to be modulated to the same frequency without interfering with each other. This mulitplexed signal will then be sent to a software defined radio [1] for use by the Time Reversal Signal Processing Scholar team. The signals will then all be sent out to demultiplexer where each signal will be reproduced to be identical to the original signal, excluding delays.

Group members are Curtis Reule and Joe Puhalla, many of you may know us as the Communications and Circuits II TA's respectively. Dr. Farden our advisor, pictured below. We have also recieved a great deal of assistance thus far from Dr. Glower, the Supreme Ruler, and Dr. Green.

The Crazy People Behind SD0621
What is the square e for?	DC Farden	Supreme Ruler!	Why am I so mean to students?

Progress Through 403 and Part of 405

This project has changed in scope quite drastically since we were first given an idea of what our project was. Originally we were going to build a 32 channel MUX/DEMUX using QAM. Since that time our project has taken on a more realistic and modular approach, where we went from having 2 extremely large PCBs to having a bunch of smaller boards. We are still being held to all of the original requirements except for number of channels. The modularized design will also allow for a much more portable system.

There are going to be 4 main boards now (2 clock boards, 1 multiplexer board, and 1 demultiplexer board). Each set of four boards will be fully capable of modulating 3 complex channels at one time. These boards are also built in a manner in which they will be easily connected to another set of boards, which allows this project to take on any number of complex channels, provided that all signals are modulated within the 0-2 MHz range.

At this time we are waiting for a set of PCBs to be delivered, ordering extra parts, and continuing with documentation.

Math, the reason we are here!

The purpose of this section is to provide a more technical representation of what the FDM using QAM actually does. For the purpose of understanding we will be investigating the effects on only one complex channel from input to output.

First we must define a few necessary signals: Let

c (t)

be a square wave and

s (t)

be a square wave 90 degrees out of phase with

c (t)

; therefore,

s (t) = j * c (t)

. These waves are in quadrature meaning they can't interfere with each other. You can also think of them in such a manner that

c (t)

is similar to a squared off cosine wave and

s (t)

is similar to a squared of sine wave. These signals will be used in the mixing stage.

Next, we must define two input signals

x 1 (t)

and

x 2 (t)

these signals will be in the audible range. (20 Hz - 20 kHz) For the purpose of this example we will Let:

(1)

x 1 (t) = c o s (1000 * 2π * t) + s i n (25000 * 2π * t)

(2)

x 2 (t) = c o s (1000 * 2π * t) + c o s (48000 * 2π * t)

x 1 (t)

and

x 2 (t)

will be sent through their own individual low pass filters. These filters band limit the signals to ensure they are in the 0-20 kHz frequency range. Let the filter be denoted as:

(3)

H l p (f) = r e c t (f / 40000)

The process of finding the output of the filters can be done using convolution; however, we will use the frequency domain and Fourier Analysis and simply multiply the signals together. Please note the following notation:

X 1 (f)

is the Fourier transform of

x 1 (t)

.

(4)

X 1 l p (f) = X 1 (f) * H l p (f)

(5)

X 2 l p (f) = X 2 (f) * H l p (f)

x 1 l p (t)

and

x 2 l p (t)

, the inverse Fourier transforms of

X 1 l p (f)

and

X 2 l p (f)

, will be modulated out to a much higher frequency by the balanced mixers and the signals

c (t)

and

s (t)

. Once again this will be done in the Fourier domain for simplicity.

(6)

X 1 m o d (f) = X 1 l p (f) * C (f)

(7)

X 2 m o d (f) = X 2 l p (f) * S (f) = X 2 l p (f) * j * C (f)

This can also be shown by use of the Real operator, which Dr. Farden often does. These signals are then summed and sent through a bandpass filter.

(8)

H b p (f) = H l p (f - f b p) + H l p (f + f b p)

(9)

X b p (f) = (X 1 m o d (f) + j * X 2 m o d (f)) * H b p (f)

This gives us theoretical perfect square waves. Where:

(10)

X b p (f) = X 1 m o d (f) * H b p (f) i + X 2 m o d (f) * H b p (f) q

Please note: Failed to parse (syntax error): _i

denotes inline and Failed to parse (syntax error): _q
denotes quadrature.

The signals are then sent to the demultiplexer stage where they are first run through bandpass filters to pick out the desired frequency, which will give us the signal

X b p (f)

again. Next, the signals are sent into the demixer, which multiplies the modulated band passed signal

X b p (f)

by another signal with the exact same carrier frequency. Using the knowledge that

c o s (φ) * c o s (θ) = 0.5 * c o s (φ + θ) + 0.5 * c o s (φ - θ)

. Which provides us with:

(11)

X 1 d e m o d (f) = X 1 m o d (f) * C (f) = X 1 (f) + X 1 (f) * C (2 * f)

(12)

X 2 d e m o d (f) = X 2 m o d (f) * S (t) = X 2 m o d (f) * j * j * C (f) = - X 2 m o d (f) * C (f) = - (X 2 (f) + X 2 (f) * C (2 * f))

Both signals are then sent through the same LPF

H l p (f)

to remove the high frequency component.

(13)

X 1 (f) = X 1 d e m o d (f) * H l p (f) = X 1 (f)

(14)

X 2 (f) = X 2 d e m o d (f) * H l p (f) = - X 2 (f)

The inline signal or

X 1 (f)

is ready to be sent out; unfortunately, we need to invert the quadrature signal or

X 2 (f)

. This is done by a simple inverting Op-Amp at the end of each quadrature signal. We then end up with our desired result of:

(15)

X 1 (f) = X 1 (f); X 2 (f) = X 2 (f)

Technical Documents

Media:UsersManual.doc

Media:FinalReport.doc

Media:FinalReportAppendix.doc

Group SD0621

From ECE Department Wiki

Contents

Group SD0621 - Frequency Division Multiplexer

Progress Through 403 and Part of 405

Math, the reason we are here!

Technical Documents

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