Arlo Clarke and Trevor LeDoyt Tufts University, Department of Electrical Engineering

12/10/2016

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.

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.

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.

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.

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)

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)

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

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.

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.

MMSE Eye Diagrams

Results

Below are the output images of our implemented system given the famous “camera man” input image.

Output Images at 10mW Noise

Output Images at 80mW Noise

Output Images at 800mW Noise

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):

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.

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)

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

Outdoor Channel Output Images (800mW Noise)