1. Arlo Clarke and Trevor LeDoyt
Tufts University, Department of Electrical Engineering
12/10/2016
2. Project Description and Overview
In this project we were asked to develop and demonstrate a working communication
simulation in Matlab. The project was designed to simulate sending an image through a noisy
channel using bipolar pulse amplitude modulation. A complete block diagram of the system is
shown below.
Modulation:
To modulate the data, we used two pulse shaping functions, the half-sine pulse and the
root raised cosine pulse. We normalized the energy in each of these pulses to ½ Joule. The two
pulse shaping functions are plotted below. Each of the pulses satisfies Nyquist criteria which
states that to reduce or eliminate ISI the pulse must be 0 at all sampling intervals. Note that the
SRRC pulse itself does not exhibit zero ISI, however, when convolved with the match filter, the
output will be a raised cosine which does exhibit a zero ISI property.
3.
4. From the frequency responses of each pulse you can see that the half-sine pulse uses
more bandwidth than the SRRC pulse. The bandwidth of the pulse is equivalent to transmission
bandwidth, therefore the SRRC has a smaller transmission bandwidth.
Effect of Changing Alpha
.3α = 0 α .8= 0
We expected the sidelobes and bandwidth to be affected by the value of . Increasingα
alpha should increase the bandwidth and also cause more sidelobe attenuation. The frequency
response plots above confirm our original intuition.
5. Effect of Changing Truncation Length (K)
K = 2 K = 6
We expected to see the bandwidth of the root raised cosine pulse to decrease as we
increased K. During our analysis we sampled the function at varying truncation values of K
ranging between 2 and 6 and were not able to see a noticeable difference.
As can be seen in the eye diagrams, the half sine does not contain any pulse overlap and
therefore is a completely open eye. The root raised cosine eye diagram shows interference
caused by the overlap from neighboring pulses. The amount of pulses an individual pulse would
overlap with was dependant on the truncation length, K.
6. Channel:
To simulate sending our data through a channel, we used a filter to apply a finite impulse
response with three delayed pulses that represent echos in the environment. The frequency,
phase and impulse responses of the channel are shown below.
Channel Output Eye Diagrams
The eye diagrams after the channel were more closed than pre-channel due to ISI caused
by the reflections in the channel. There was a small opening on the innermost eye. Note that the
raised cosine eye diagram is shown over two bit intervals for a clearer picture as to how the
pulses interfere.
7. Noise
Channel Output Eye Diagrams with Added Noise
The eye diagrams of the noisy channel output were distorted due to variations in each
pulse caused by the AWGN. It can be observed that the innermost eye on these diagrams is now
completely closed. Again, note that the raised cosine eye diagram is shown over two bit intervals
for a clearer picture as to how the pulses interfere.
Matched Filter
The frequency, phase and impulse response of the matched filters were the same as the
pulse’s because both of the pulses were symmetric.
Frequency and Phase response of Matched Filter (Half-sine Pulse)
8. Impulse Response of Matched Filter (Half-Sine Pulse)
Frequency and Phase Response of the Matched Filter (SRRC Pulse)
Impulse Response of the Matched Filter (SRRC Pulse)
9. Matched Filter Output Eye Diagrams
The eye diagrams show that the optimum sampling point is at time 0, which in our case
corresponds to 32 samples (one bit interval). Note that after the output of the matched filter a bit
spans 64 samples.
Equalization
We implemented two equalization filters, a Zero-forcing filter and Minimum mean square
error (MMSE) filter. The zero forcing filter was implemented as the inverse response of the
channel while the MMSE filter was implemented to minimize the error energy as
conj(H)/(mag(H)^2 + 2*Pnoise).
Frequency and Impulse Response of Zero Forcing Filter
10. Impulse Response of Zero Forcing Filter
The frequency response of the zero forcing filter is the inverse of the channel. This is how
it is able to essentially undo the effects of the channel, but it is not designed to handle the effects
noise has on the data. As can be seen in the impulse response of the ZF filter, the channel inverse
is stable, however, the stability of the inverse of the channel can’t be generally assumed.
ZF Eye Diagrams
From the eye diagram you can see how the zero forcing filter was able to eliminate most of the ISI
and any remaining interference is minimal at the sampling point.
11. MMSE Frequency and Impulse Response
Frequency and Phase Response of MMSE filter
Impulse Response of MMSE filter
As can be seen, the MMSE impulse response converges to zero much quicker than the ZF filter.
16. Analysis
The critical SNR of the ZF process was ~22dB while the critical SNR of the MMSE process
was ~2dB. This follows due to the nature of MMSE as a filter that minimizes error energy.
The transmission process was not image dependant; that is, the same amount of error
was visible in two separate images when the same amount of noise power was injected into the
signal. This can be observed in the two images sets below (both had noise levels of 80mW):
17. The half sine pulse satisfies the nyquist criterion, but the square root raised cosine (SRRC)
does not. The ISI due to pulse overlap is removed by the matched filter because the time domain
convolution of the matched filter SRRC with the SRRC pulse itself will output a true root raised
cosine. However, we still expected ISI to be present after the matched filter due to the ISI
generated by the channel. The equalization filters will reduce or eliminate the remaining channel
ISI. Our expectations were confirmed during implementation when we found that ISI was still
present at the sampling point after the matched filter.
The half sine pulse had a larger bandwidth, however, the only noise associated with error
is the noise at the sampling point, decoupling error rate from bandwidth. The SRRC pulse has
signal overlap which could cause errors in detection and sampling but that is mostly removed by
the matched filter. These points considered, we expected the root raised cosine to have a slight
advantage in our system, even if there would be no visually noticeable differences in error rate.
Although we didn’t do any thorough calculations to determine the actual bit error
associated with each of the pulses, we did test each pulse using the same noise energy and pulse
energies and found that each of the pulses performed equally well. This is evidenced by all of the
image sets in this report, as half sine and SRRC pulses performed equally well at the same noise
level.
18. We tested the performance of our system using two additional channels. An outdoor
channel, with a length of 25 taps and an indoor channel, with a shorter length of 7 taps. The
outdoor channel took significantly longer to process and we were not able to recover our image
using the zero forcing filter. Because MMSE was still able to recover the image, we expected the
zero forcing filter to be unstable. By checking the impulse response of the zero forcing filter we
confirmed that it was unstable.
Frequency, Phase and Impulse response plots of Outdoor Channel
Output Images for Outdoor Channel (20mW Noise)
19. Unstable Impulse Response of the Channel Inverse (ZF Filter) for the Outdoor Channel
We tested the indoor channel at different noise levels and found that the zero forcing
filter outperformed the MMSE filter.
Frequency, Phase and Impulse Response Plots of Indoor Channel
