ECE 3500: Fundamentals of Signals and Systems (Fall 2015) Lab 4: Binary Phase-Shift Keying Modulation and Demodulation

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1 ECE 500: Fundamentals of Signals and Systems (Fall 2015) Lab 4: Binary Phase-Shift Keying Modulation and Demodulation Files necessary to complete this assignment: none Deliverables Due: Before Dec. 18th Introduction Modulation is an important component of communication systems. Modulation is the process of converting a baseband, low-frequency signal into a signal with high frequency components so that we can transfer information across a physical communication channel. We use modulation to transmit data over air, across outer-space, through wires, and in fiber optics. There are many types of modulation schemes. Each have a variety of advantages and disadvantages. In this lab, we fill focus on a digital communications scheme known as phase-shift keying (PSK) modulation. This is a standard scheme for wireless LAN IEEE communications systems 1. Binary phase-shift keying (BPSK) is the simplest type of phase-shift keying modulation. To implement BPSK, we assume the message (the baseband signal) is binary, i.e., it contains only 1 or 1 values. A information bit at time t controls, or modulates, the phase of the carrier cos(ω c t). If the bit is 1, we transmit cos(ω c t+0). If the information bit is 1, we transmit cos(ω c t+π). In this lab, we will simulate a binary phase-shift keying receiver system in MATLAB. In Part 1 and Part 2 of the lab, we will explore and simulate modulation/demodulation of the BPSK signal when our communication channel is ideal. In Part of the lab, we will then study the effects of channel noise and other errors at the receiver. This lab assignment has three learning objectives: 1. Understand the binary phase-shift keying modulation and demodulation process 2. Simulate binary phase-shift keying modulation / demodulation in MATLAB. Explore the effects of realistic errors on binary phase-shift keying systems Deliverables This lab assignment has two deliverables: 1. One (1) lab report (see the Lab Report Guidelines on how to write it) 2. One (1) demonstration of your working BPSK simulation to the lab instructor Please submit a hard or electronic copy of deliverables to your lab instructor before Dec. 18th. 1 Sanjeev Kumar, BER Improvement of Wireless LAN IEEE Standard Using Wavelet Packet Transforms, Journal of Communication Technology, vol., no.,

2 Part 1: Design a BPSK transceiver BPSK Modulation: We begin with deriving BPSK signals as well as the modulation and demodulation process. We represent a sequence of bits (each with value -1 or +1) as a discrete signal b[n] such that b[1] is the first bit transmitted, b[2] is the second bit transmitted, etc. We can convert these bits into a continuous-time representation by holding the value for a period of T seconds. Therefore, our continuous-time, baseband signal can be represented by b(t) = b[1], 0 t < T b[2], T t < 2T. b[n], (N 1)T t < NT This represents the continuous-time message that we want to transmit. From this information, we can define the BPSK modulated signal x(t) as x(t) = b(t)cos(ω c t+φ), (2) where ω c and φ represent the frequency and phase of the carrier, respectively. If φ = 0, the signal will have a phase of 0 when b(t) = 1 and a phase of π when b(t) = 1. To help understand how BPSK works, assume the Fourier transform of b(t) is B(ω) with a bandwidth ω B (i.e., has non-zero values for ω B ω ω B ). Determine the Fourier transform of x(t). (1) BPSK Demodulation: Now we discuss the demodulation process. In practice, a receiver will not have perfect knowledge of a carrier s frequency or phase. In this lab, we will assume that both the carrier frequency and phase are known. Although, in the last part of the lab, we will explore how errors in these assumed frequency and phase affect our results. When the frequency and phase are known, the demodulation process on x(t) is as follows: 1. Multiply x(t) with cos(ω c t+φ) such that x 1 (t) = x(t)cos(ω c t+φ) 2. Apply a low-pass filter h(t) to x 1 (t) to remove the higher frequency interference. The result, b(t) = x 1 (t) h(t), is an estimate of b(t). To help understand how BPSK works, determine the Fourier transform of b(t) based on the Fourier transform of x(t) that was determined during our discussion of BPSK modulation. Message Retrieval: BPSK modulation is used to transmit a message at a high frequency. BPSK demodulation returns the received high frequency signal to low frequencies. After demodulation, the last step is to retrieve/decode the original discrete-time message b[1],b[2],b[],... from the demodulated continuous-time signal b(t). Consider how we might be able to accomplish this we will revisit this question in Part #2. 2

3 Part 2: Simulate a BPSK system in MATLAB Generate a bit sequence / message: For testing purposes, we will consider a random discrete sequence of bits with values of +1 or -1. This sequence represents some arbitrary message that we want to transmit with BPSK. To generate these bits, first use the MATLAB command >> c = randn(25,1); to generate a vector with 25 randomly chosen values ranging from to. The random numbers are taken form a normal (or Gaussian) distribution. Now develop a command or set of commands that will convert the vector c into vector b such that b[n] = { 1 ; c[n] 0 1 ; c[n] < 0. In the above expression, b[n] represents the value in vector b at index n and c[n] represents the value in vector c at index n. The resulting vector b should only contain +1 values and 1 values. Modulate your bit sequence: Although we have been discussing continuous-time modulation, we will simulate the system in discrete-time with MATLAB. For this simulation, assume 1. A bit duration of T = s 2. A carrier frequency of f c = 100,000 Hz. A carrier phase of φ = 0 radians 4. A sampling frequency of f s = 10 6 samples/s (sampling period of Ts = 10 6 s/sample) Use these values to determine the number of samples per bit and total number of samples of your transmitted signals. Now develop a MATLAB function function [x, btilde] = BPSK(b,fc,phi,T,Ts) that generates the sampled values of the modulated signal x(t) given the sequence of bits b, the carrier frequency fc, the phase of the carrier phi, the bit duration T, and the sampling period Ts. The BPSK function has two outputs: x and btilde. The vector x represents the modulated, transmitted signal x(t). The vector btilde represents the continuous-time sequence of bits in b(t). Use the pwelch command in MATLAB to estimate the spectrum of btilde and x to make sure that the modulation behaves as you derived in Part 1. Demodulate your transmitted signal: In the previous subsection, we modulated our binary message. Now we will demodulate it. Write a MATLAB function function bhat = DBPSK(x,fc,phi,T,Ts) that processes the modulated BPSK signal x to recreate (or estimate) the baseband signal b(t). You will need to design the lowpass filter to complete the demodulation. To design this filter, use the MATLAB function

4 h = fir1(n,wn). This command designs an N-th order lowpass filter h(t) with a cutoff frequency of Wn. Note that in MATLAB, a frequency of Wn = 1 corresponds to half the sampling frequency. Therefore, the parameter Wn should be selected with this oddity in mind. The number of coefficients N (i.e., the length of the filter s impulse response signal) will determine how steeply the magnitude response of the filter goes down in the stop band. Since we have not studied discrete-time filters, you should determine the parameter N by trial and error. Use the command freqz(h,1,1024,1/ts) to plot the magnitude and phase response of the filter your design. Once the filter is designed, use bhat = conv(x1,h, same ) to generate bhat from x1. The vectors x1 and bhat represent x 1 (t) and b(t), respectively, from Part 1. Again, use pwelch to make sure that bhat is as theory predicted. Estimate your message: Now design and implement a program in MATLAB that determines the original discrete-time message that was sent from the simulated continuous-time signal bhat. This is should not be too difficult and there are multiple options. Test your three components to ensure you can successfully modulate, demodulate, and decode a message. Once you have made sure the transmitter, receiver, and estimator are working as expected, you are ready to explore imperfections that are unavoidable in the real world. 4

5 Part : Study your design in realistic conditions Channel noise: In real life, we transmit data across noisy channels or environments. We often model the noise as additive white Gaussian noise. This is a common model for background noise in electronics. An additive white Gaussian noise corrupted signal x(t) can be expressed as In MATLAB we can simulate the above with the command xr = x + sigma*randn(l,1) x r (t) = x(t)+η(t). () where L contains the number of samples in xr and x. The constant sigma is the standard deviation of the noise. Note that you may need to use randn(l,1) or randn(1,l) depending on whether xr and x are column vectors or row vectors, respectively. Develop a MATLAB program that repeats the modulation and demodulation process from Part 2, but with a few differences. Now use 100 bits and perform demodulation with the noisy received signal as described above. Vary sigma from 0 to 50 in steps of 0.1. For each value of sigma, compute the communication system s bit error rate (BER), defined by BER = Number of bits incorrectly estimated Total number of bits Plot the bit error rate (BER) as a function of sigma, the standard deviation.. (4) Carrier frequency error: Repeat the above analysis for when the carrier frequency fc of the demodulator (DBPSK) is different from the carrier frequency of the modulator (BPSK). Vary this difference from 0 Hz to 100 Hz in increments of 0.5 Hz. Plot the resulting bit error rate (BER) as a function of the frequency error. Phase error: Repeat the above analysis for when the carrier phase phi of the demodulator(dbpsk) is different from the carrier phase of the modulator (BPSK). Vary this difference from 0 radians to π radians in increments of π/100 radians. Plot the resulting bit error rate (BER) as a function of phase error. Lab Report Guidelines / Template The lab report template can be found on the course website: Please follow the guidelines in this document and the grading rubric on the next page. Every section of the lab should be written in full sentences and paragraphs with proper English. Note: In your report, do not use any text that is copied and pasted from these lab instructions. 5

6 Lab #4 Grading Rubric Section Requirement Scored Possible Points Structure (1 points) Uses the pre-defined template and uses the correct formatting. 2 Report is less than 5 pages (without figures and tables) 5 Includes an appropriate title, author name, lab section, and address 1 Includes an Abstract 1 Includes an Introduction & Motivation Section 1 Includes an Experimental & Analytical Methods Section 1 Includes a Results and Discussion Section 1 Includes a Conclusions Section 1 Abstract (7 points) Provide: Abstract is less than 120 words Provide: Describes motivation of report 2 Provide: Describes methods and results in the report 2 Introduction & Motivation (10 points) Provide: An overview of the project (i.e., what are you doing?) Provide: Includes the report motivation for readers (i.e., why do I care) Experimental & Analytical Methods (24 points) Provide: Summary of experimental and analytical methods you use (i.e., how are you doing this in 2 sentences) Provide: Briefly summarizes your results (i.e., did you succeed or fail by how much?) Answer: How did you create the transmitter / modulator? Describe using minimal code. Use math (which is more universal) instead. Write this so that we could replicate your results. Answer: How did you create the receiver / demodulator? Describe using minimal code. Use math (which is more universal) instead. Write this so that we could replicate your results. Answer: What type of signal(s)/data do you need to complete this simulation / analysis of the BPSK receiver? How did you generate this data? Write this so that we could replicate your results. 2 2 Answer: What are your generated signals simulating? Answer: After demodulation, how do you determine your message? Write this so that we could replicate your results. Answer: Describe your realistic condition analysis. Specifically, describe the channel noise analysis. What does channel noise represent (i.e., why investigate this?)?

7 Answer: Describe your realistic condition analysis. Specifically, describe the carrier frequency error analysis. What does carrier frequency error represent (i.e., why investigate this?)? Answer: Describe your realistic condition analysis. Specifically, describe the phase error analysis. What does phase error represent (i.e., why investigate this?)? Results & Discussion (24 points) Conclusions (8 points) References (4 points) Deliverables (10 points) Answer: Describe how well bhat recover s btilde without any error. If there are differences, explain why they occur. Include: One or two plots comparing btilde and bhat. Zoom into the plots for better comparison. Remember to label you axes and include units. Answer: Describe how well bhat recover s btilde without with channel noise. If there are differences, explain why they occur. Include: Plot of bit error rate (BER) versus channel noise standard deviation. Cite this in the text. Describe why the plot looks like this. Remember to label you axes and include units. Answer: Describe how well bhat recover s btilde without with carrier frequency error. If there are differences, explain why they occur. Include: Plot of bit error rate (BER) versus carrier frequency error. Cite this in the text. Describe why the plot looks like this. Remember to label you axes and include units. Answer: Describe how well bhat recover s btilde without with carrier phase error. If there are differences, explain why they occur. Include: Plot of bit error rate (BER) versus phase error. Cite this in the text. Describe why the plot looks like this. Remember to label you axes and include units. What did you demonstrate? Was your experiment a success? 2 Could your methodology and results be improved? If so, how? How could this initial experiment be expanded to larger, bigger engineering problems? What potential are you demonstrating through this project? References are used (or not used) appropriately for the content of the report Demonstrate your working BPSK system to your lab TA Total

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