Project Title: e-laboratories for Physics and Engineering Education Tempus Project: contract # 517102-TEMPUS-1-2011-1-SE-TEMPUS-JPCR 1. Experiment Category: Electrical Engineering >> Communications 2. Experiment Name: 3. Date and Issue number: 14/9/2014, v8.0 4. Instructor Name: Prof. Hani Ghali Eng. Sameh Osama Pulse Code Modulation (PCM) 5. Institution: The British University in Egypt (BUE) 1 Page 17
Table of Contents 1. Experiment Overview 3 2. Intended Learning Outcomes (ILOs) 3 3. Introduction: Theoretical Background 3 3. I Sampling of Analog Signals 4 3. II Quantization 7 3. III Encoding 9 3. IV PCM Demodulation 9 3. IV. I Digital to Analog Conversion 9 3. IV. II Smoothing Filter 9 4. References 10 5. Relation to Course Contents and Topics 10 6. Needed Equipment 10 7. Experiment Output 11 7. I Task (1) 11 7. II Task (2) 11 7. III Task (3) 11 7. IV Task (3) 11 8. Experiment Steps 12 8. I Task (1): Sampling 12 8. II Task (2): Quantization 13 8. III Task (3): Encoding 14 8. IV Task (4): Decoding & Filtering 15 9. Interface Instructions/Help 16 2 Page 17
Pulse Code Modulation (PCM) 1. Experiment Overview The main objective of this experiment is to familiarize students with the Pulse Code Modulation (PCM) technique. Students will be able to practice the three main concepts; 1) sampling, 2) quantization and 3) encoding. The experiment is associated with two courses: A. Communications: taught in semester one and two for second year students in Electrical Engineering Department B. Principles of Digital Communication (1): taught in semester one and two for third year students in Electrical Engineering Department 2. Intended Learning Outcomes (ILOs) Upon completion of this experiment, students should be able to: 1. Apply the sampling process 2. Implement both the quantization and encoding processes 3. Explain the concepts of analog to digital and digital to analog conversions 3. Introduction: Theoretical Background Pulse Code Modulation (PCM) is a modulation technique associated with the transmission of an analog input signal only in discrete time. The analog signal is firstly sampled in time-domain. These analog samples will then be converted to another set of samples but with predefined values through a process called quantization. Finally, the set of quantized samples will then be converted into a code (digits) which is the final form of transmission. In the receiving end, the received code is passed through a low-pass filter to recover the original signal. The PCM block diagram and process flow is shown in Fig. 1. x t x s t x q t Sampling Quantization Analog Signal f s 2f x Fig. 1: PCM Block Diagram Coding Digital Signal 3 Page 17
3. I Sampling of Analog Signals Sampling is defined as the process of conversion of an analog signal into a discrete-time sampled signal. This is achieved through the multiplication of the time domain analogue signal "x t " by a periodic train of impulses "s t " [sampling function] as shown in Fig. 2. The period "T s " is called the sampling period and "F s =1/T s " is the sampling frequency, where: s t = δ t nt s n=, T s = 1/F s, ω s = 2 π F s [1] The sampled signal "x s t " is given by; x s t = s t. x t = x nt s δ t nt s n= [2] Sampling function s t x t Input analog signal x s t Output sampled signal a) Block diagram b) Signals involved in the sampling process Fig. 2: Sampling Process 4 Page 17
From the Fourier multiplication property, X s f is the convolution of X f and S f, therefore: X s f = Knowing that; X θ S[ f θ ] dθ S f = 1 T s δ f kf s k= Since, the convolution with an impulse simply shifts the signal; [eq. 3] could be expressed as; X s f = I T s X [ f kf s ] k= [3] [4] [5] The signal X s f is a periodic function of f consisting of superposition of shifted replicas of X f scaled by 1/T s " as shown in Fig. 3. Different spectrum components are shown in Fig. 3 as follows; the signal X f has a maximum frequency of 100 Hz as shown in Fig. 3(a), the sampling frequency is 250 Hz as shown in Fig. 3(b), and the result of sampling is shown in Fig. 3(c), which consists of shifted replicas located at multiple of the sampling frequency; 0 Hz, 250 Hz, and 500 Hz. Fig. 3(a) Fig. 3(b) Fig. 3(c) 5 Page 17 Fig. 3: Spectrum of the message signal, sampling signal and sampled signal
For the case f s < 2 f x, where f x is the highest frequency in the message signal, an overlapping in the frequency domain between the shifted replicas occurs as shown in Fig. 4. This overlapping is called Aliasing which affects the reconstruction of the original message signal. In Fig. 4, the signal X f has a maximum frequency of 100 Hz as shown in Fig. 4(a), the sampling frequency is now 150 Hz as shown in Fig. 4(b), and the result of sampling is shown in Fig. 4(c), which consists of shifted replicas located at multiple of the sampling frequency; 0Hz, 150Hz, and 300Hz. The overlapping is clear and is due to the fact that in this case the sampling frequency 150Hz is less than twice the maximum of the highest frequency in the message signal 200Hz. To avoid such phenomena, the sampling frequency should be equal to or greater than double the maximum signal frequency. Fig. 4(a) Fig. 4(b) Fig. 4(c) Fig. 4: Aliasing Phenomena 6 Page 17
3. II Quantization After the sampling process takes place, the signal appears as analog samples which can, theoretically, still take any value corresponding to the value of the original analog signal at the sample instance. Consequently, there are infinitely possible values for each sample. However, to be able to represent (encode) each sample value from such a continuous range into a finite number of digits (code), the possible values of the samples should be limited to finite number. This is achieved through a process named quantization as shown in Fig. 5. In this process, a fixed number of possible values for the samples, called quantization levels "M", is defined and should be in the form; M = 2 n [6] Where "n" an integer whose value is equal to the number of encoded bits Quantized signal Original signal Fig. 5: Quantization of the message signal The number of quantization levels "M" is determined by the analog-to-digital converter characteristics, specifically, the analog input voltage range V min and V max and the number "n" of corresponding encoding bits. The quantization levels are separated by step size given by; = V max V min 2 n [7] Any given analog sample will be approximated to the closest quantization level producing a maximum quantization error of " /2". 7 Page 17
The quantization error can be considered as a noise introduced to the original analog samples before being quantized. Fig. 6(a) shows both the original signal (blue) and the quantized signal (yellow), while Fig. 6(b) shows the difference between these two signals, which is the noise added to the original samples. This quantization noise has a power given by: N = 2 12 [8] Quantized signal Original signal Fig. 6(a) Fig. 6(b) Fig. 6: Original signal, quantized signal and the difference (quantization noise/error). 8 Page 17
3. III Encoding The quantized of samples having possible "M" values, which are discrete in time and amplitude are then converted into a binary signal (2 levels) by using pulse coding. The code is formed by assigning a binary representation of the decimal number to each of the levels from "0" to"m 1". In a "M = 32" signal for example, using [eq. 6] the required number of bits is "n = 5", while for the case of "M = 8" the required number of bits is "n = 3". For this case, the code for level "0" is "000", and the code "111" represents level "7". When this pulse-coded signal is transmitted, "n" binary digits (or bits) are sent for each sample "T s " seconds. 3. IV PCM Demodulation 3. IV. I Digital to Analog Conversion During the transmission of the PCM signal, noise interference is unavoidable. Consequently, before the PCM signal is processed for reconstruction, a comparator is used to recover the received signal to the original sent series of bits (pulses). Then the signal will pass through an n- bits decoder (normally should be a D/A converter) which will convert the received digital signal to the original quantized values. Finally a smoothing filter is used to reconstruct the original analog signal from the quantized one. Fig. 7 shows the block diagram of a PCM demodulator. x s t Decoder x q t Low Pass Filter Digital Signal Fig. 7: PCM Demodulator Block Diagram 3. IV. II Smoothing Filter The quantization value not only includes the original signal, but also includes a lot of high frequency harmonics; therefore, a low-pass filter is required to remove the unwanted harmonics at the receiving end. 9 Page 17
4. References [1] B. P. Lathi, Modern Digital and Analog Communication Systems (Oxford Series in Electrical and Computer Engineering), Oxford University Press, 4 th edition, ISBN: 978-0195331455, 2009. [2] S. Haykin, Communication Systems, Wiley; 5th edition, ISBN: 978-0471697909, 2009. [3] Alan V. Oppenheim, Alan S. Willsky, and S. Hamid, Signals and Systems, Prentice Hall; 2nd edition, ISBN: 978-0138147570, 1996. [4] Sophocles Orfanidis, Introduction to Signal Processing, Prentice Hall, ISBN: 978-0132091725, 1995. [5] Emmanuel Ifeachor and Barrie Jervis, Digital Signal Processing: A Practical Approach, Prentice Hall; 2nd edition, ISBN: 978-0201596199, 2001. 5. Relation to Course Contents and Topics This experiment is designed to cover the following topics in the two previously mentioned courses: 1. Sampling 2. Quantization 3. Encoding 4. Decoding & Filtering 6. Needed Equipment No hardware equipment are needed 10 Page 17
7. Experiment Output The followings are the expected output that student should get by the end of the experiment: 7. I Task (1) With a given analog time domain signal, the student should perform ideal sampling through multiplying this given signal by a train of impulses producing a discrete time signal. 7. II Task (2) In this part, the student should perform quantization for a discrete time domain signal. Student should detect change in the quantization error as a result of changing the number of levels. In addition to the simulation, the student should calculate analytically the quantization error for the following cases: a. M=2 b. M=4 c. M=8 d. M=16 e. M=32 Finally, the student should observe that as the number of levels increases the quantization error decreases which reflect on the ability to reconstruct the original signal in the receiver. 7. III Task (3) In this section, the student should perform the encoding process for a discrete signal. Student should observe the effect of the number of levels chosen in the previous part on the output number of bits. 7. IV Task (4) In this task, the student should perform the decoding process for the stream of binary data. Student should observe the effect of the something filter cut-off frequency on the reconstructed signal. 11 Page 17
8. Experiment Steps This experiment is divided into four tasks: 1. Sampling 2. Quantization 3. Encoding 4. Decoding & Filtering 8. I Task (1): Sampling In this task, the student selects will be able to change the amplitude and frequency of a sinusoidal time-domain message signal. In addition, the student will be also able to change the frequency of the sampling signal and then sample the message signal through multiplying it by the sampling signal. By changing the sampling period, student will be able to observe the effect of varying the sampling period on the output number of samples. Steps 1. Adjust the sampling frequency to be around 30 MHz 2. Adjust the input signal frequency to 10 MHz 3. Determine the number of samples per period of the sampled signal 4. Change the sampling frequency from 30 to 100 MHz with step 10 MHz and repeat step 3. 12 Page 17
8. II Task (2): Quantization In this task, the student will be able to vary the number of levels of the quantizer and observe its effect on the quantized signal. In addition, the student will observe the quantization error and compare it with that calculated analytically. Steps 1. Adjust the sampling frequency to be around 30 MHz 2. Adjust the number of levels to be 2 3. Determine the quantization step size and the quantization noise 4. Change the number of levels to be 4, 8, 16 and 32, and in each repeat step 3 5. Plot a graph for the number of bits vs. the quantization noise 6. What is the main disadvantage of increasing the number of levels? 13 Page 17
8. III Task (3): Encoding In this task, the student will get experience in the use of encoder and the analog to digital conversion process. Steps 1. Adjust the sampling frequency to be around 30 MHz 2. Set the number of bits to be 32 and write the binary equivalent of the output digital stream 3. Change the number of bits and study its effect on the digital output 14 Page 17
8. IV Task (4): Decoding & Filtering In this task, the student will get experience about the decoding process for a stream of binary data. The student should observe the effect of the something filter cut-off frequency in constricting the signal. Steps 1. Vary the cut-off frequency of the low pass filter and detect its effect on the reconstructed signal 2. Compare this frequency with that of the input sinusoidal signal 3. Vary the number of level and observe its effect on the reconstructed signal. 15 Page 17
9. Interface Instructions/Help The left knob is used to adjust the amplitude of the input message signal The right knob is used to adjust the frequency of the input message signal This knob is used to adjust the frequency of the sampling signal The left selector is used to choose whether to display the input message signal in time domain or not The right selector is used to choose whether to display output signal or not The left knob is used to adjust the value of the vertical display in volts/division for the time domain plot The right knob is used to adjust the value of microseconds/division for the time (horizontal) axis This knob is used to adjust the number of quantization levels 16 Page 17
17 Page 17 This knob is used to adjust the cut off frequency of the used Low Pass Filter (LPF)