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1. Module 02: Lecture Slides DIGITAL COMMUNICATION  (BEC503) SOMESH B S Assistant Professor AY 2025-26 Department of Electronics & Communication Engineering (Accredited by NBA) Atria Institute of Technology (An Autonomous Institution) Anandanagar , Bengaluru- 560 024 Accredited by NAAC with A ++ Grade

2. Module 1 Digital Modulation Techniques: Phase shift Keying techniques using coherent detection: generation, detection and error probabilities of BPSK and QPSK, M– ary PSK, M– ary QAM. Frequency shift keying techniques using Coherent detection: BFSK generation, detection and error probability. BFSK using Noncoherent Detection, Differential Phase Shift Keying. Simon Haykin , “Digital Communication Systems”, John Wiley & sons, 2014, ISBN 978-81- 265-4231-4.

3. Fundamentals of Digital Modulation The process of varying one or more properties of a carrier signal with respect to message signal is called modulation. In digital modulation the message signal will be binary data, which is to be carried by an analog signal for transmission over channel. Data transmission can be classified as Baseband transmission and Passband transmission.

4. Fundamentals of Digital Modulation

5. Fundamentals of Digital Modulation Baseband Transmission: Digital data is transmitted over the channel directly. Suitable for short range communication. Passband Transmission: Digital data modulates high frequency sinusoidal carrier for transmission over a wireless channel. Suitable for long range communication. Binary signaling involves switching or keying the amplitude, phase or frequency of carrier in accordance with digital data.

6. Fundamentals of Digital Modulation Binary signaling involves switching or keying the amplitude, phase or frequency of carrier in accordance with digital data. The receiver can be either a coherent or non-coherent detection. Coherent Detection: The local carrier generated at the receiver will be in phase and frequency synchronization with the carrier generated at the transmitter. Also called synchronous detection. The probability of error is greatly reduced. Coherent detection is performed as follows: Correlation of received signal with carrier Decision making based in threshold values

7. Fundamentals of Digital Modulation Non-Coherent Detection: The carrier at the receiver need not be in phase with carrier at the transmitter Simpler method but has higher probability of error.

8. Binary Phase Shift Keying ( BPSK)

9. Binary Phase Shift Keying ( BPSK) In BPSK symbols 1 and 0 modulate the phase of the carrier. It transmits 1 bit/symbol i.e., n=1 or M= 2 1 =2 Hence 2 symbols 1 and 0 are denoted by signals s 1 (t) and s 2 (t) respectively i.e., and When the symbol changes from 1 to 0 or vice versa the phase of the carrier is shifted by 180 ( ). Let E b denote the energy of the symbol and T b denote the bit duration.  

10. Binary Phase Shift Keying ( BPSK) Let E b denote the energy of the symbol and T b denote the bit duration. E b = = = = = = If f c is high and T b contains integer number of carrier cycles, then sin〖(4𝜋𝑓 𝑐 𝑇 𝑏 )〗 ≅0. This implies that E b = 𝑇 𝑏 or A = . The signals can be represented now as: where E b is the transmitted energy per bit and T b is the bit duration.  

11. BPSK: Signal Space Diagram In BPSK symbols 1 and 0 are represented by s 1 (t) and s 2 (t) respectively where, --------------------> (1) -------------------> (2) When the symbol changes from 1 to 0 or vice versa the phase of the carrier is shifted by 180 ( ). Signals (1) and (2) are antipodal signals. From equation (1) and (2) there is only one orthonormal basis function (ONBF): = = ; ------> (3) ; --------------------- -----------------> (4) ; -----------------------------------> (5)  

12. BPSK: Signal Space Diagram ∴ BPSK is having a signal space of 1 dimension with a signal constellation of 2 message points (M=2). The coordinates to represent these 2 message points are given by: s 11 = = || from equation (4) = s 11 = --------------------------------------------------------------------------------> (6) Similarly s 21 = - -----------------------------------------------------------------------> (7)  

13. BPSK: Signal Space Diagram

14. Binary Phase Shift Keying ( BPSK)

15. BPSK: Generation BPSK generator consists of 2 components: Polar NRZ level encoder: The block converts symbol 1 and 0 of the input binary sequence by amplitudes +√(𝐸 𝑏 ) 𝑎𝑛𝑑−√(𝐸 𝑏 ) 𝑟𝑒𝑠𝑝𝑒𝑐𝑡𝑖𝑣𝑙𝑒𝑦. Product Modulator: The output of polar NRZ level encoder is multiplied by the ONBF ɸ 1 (𝑡) i.e., ɸ 1 (𝑡) acts as a sinusoidal carrier for BPSK signal.

16. BPSK: Coherent Detection Coherent BPSK Detection has 2 basic components: Correlator: Correlates the received signal s(t) with the ONBF ɸ 1 (𝑡) Decision Device: Compares the correlator output against 0 threshold assuming binary symbols 1 and 0 are equiprobable. If x 1 > 0 then decision is made in favor of symbol ‘1’. If x 1 < 0 then decision is made in favor of symbol ‘0’.

17. BPSK: Probability of error for coherent detection Assume symbols 1 and 0 are transmitted with equal probability. If the received set of points resides close to s 11 then it corresponds to symbol 1 transmission. If the received set of points resides close to s 21 then it corresponds to symbol 0 transmission. These 2 decision regions are marked Z 1 and Z 2 respectively on the signal constellation diagram. The decision rule is now simply to decide that signal 𝑠 1 (t) is transmitted if the received signal point falls in Z 1 region and to decide that signal 𝑠 2 (t) is transmitted if the received signal point falls in Z 2 region.

18. BPSK: Probability of error for coherent detection Two kinds of erroneous decisions can be made: Error of the first kind: Signal 𝑠 2 (t) is transmitted and due to noise, the received signals falls in Z 1 region and receiver decides in favor of signal 𝑠 1 (t). Error of the second kind: Signal 𝑠 1 (t) is transmitted and due to noise, the received signals falls in Z2 region and receiver decides in favor of signal 𝑠 2 (t).

19. BPSK: Probability of error for coherent detection Calculation of probability for making error of first kind: Let x(t) be the received signal and x(t) = s(t) + w(t) -----------------------------------------> (1) where w(t) is the white Gaussian noise with 0 mean and power spectral density of . If s 2 (t) / symbol ‘0’ was transmitted, then the output of the correlator is given by: x1 = = = + || but = = + || w1= + w 1 x 1 + w 1  

20. BPSK: Probability of error for coherent detection Mean of the random variable x 1 Variance of the random variable x1 = E [x 1 ] = E [ + w 1 ] = E [ ] + E [w 1 ] = - + 0 || mean of a constant = constant and mean of AWGN is 0. = - = var [x 1 ] = var [ + w 1 ] = var [ ] + var [w 1 ] = 0 + || variance of a constant = 0 and variance of AWGN is . = Mean of the random variable x 1 Variance of the random variable x1

21. BPSK: Probability of error for coherent detection Calculation of probability for making error of first kind: Conditional probability density function when symbol 0 / s2(t) is transmitted is given by: = Substituting for and we get: = and = Let P e (0) be the probability of deciding in favor of symbol 1 when symbol 0 is transmitted. i.e., P e (0) = P[ >0 | symbol 0 is transmitted]; region Z 1  

22. BPSK: Probability of error for coherent detection Calculation of probability for making error of first kind: Let z = , when x = 0 then z = and when x = then z = and differentiating we get dz = or dx = dz dz dz Definition of complimentary error function erfc (u) = du = erfc (z) = erfc ( ) Similarly, we can calculate the probability of error for second kind, i.e., symbol 1 / s 1 (t) is transmitted but receiver decides in favor of symbol 0 / s 2 (t) i.e., = erfc ( ).  

23. BPSK: Probability of error for coherent detection Calculation of probability for making error of first kind: Let the probability of transmitting 1 and 0 be equiprobable i.e., = = . The average probability of error is given by = + ( erfc ( )) + ( erfc ( )) = erfc ( ) Relation between erfc (x) and Q(x) is Q(x) = erfc ( ). probability of error of coherent BPSK detector is given by = Q( )  

24. Quadrature Phase Shift Keying [QPSK]

25. Quadrature Phase Shift Keying [QPSK] In BPSK, we transmit 1 bit/symbol [i.e., n=1; M=2], but in QPSK we transmit 2 bits/symbol which helps in bandwidth conservation, thus the name bandwidth conserving modulation scheme. 2 bits/symbol can have 4 possible combinations [i.e., n=2; M=2 2 =4] of message symbols = 11, 01,00,10 To transmit 2 bits/symbol we need 4 phase shifts [ quadri phase shift] = = radians. There is a separation of phase angle between the carrier phases. In QPSK, the phase of the carrier takes one of four equally spaced values/ phases such as: = ; + = + =  

26. Quadrature Phase Shift Keying [QPSK]

27. QPSK: Signal space representation As with binary PSK, information about the message symbols in QPSK is contained in the carrier phase. In particular, the phase of the carrier takes on one of four equally spaced values, such as . For this set of values, we may define the transmitted signal as -------------------------------------> (1) where E is the transmitted signal energy per symbol and T is the symbol duration. The carrier frequency f c equals nc / T for some fixed integer n c . Each possible value of the phase corresponds to a unique dibit (i.e., pair of bits). Thus, for example, we may choose the foregoing set of phase values to represent the Gray-encoded set of dibits , 10, 00, 01, and 11.  

28. QPSK: Signal space representation Expanding equation(1) to redefine the transmitted signal in the canonical form: where i = 1, 2, 3, 4. Based on this representation, we make two observations: There are two orthonormal basis functions, defined by a pair of quadrature carriers:

29. QPSK: Signal space representation There are four message points and associated signal vectoring using equation (1) as follows: for dibit 11 for dibit 01 for dibit 00 for dibit 10  

30. QPSK: Signal space representation The four message points can be defined by 2 dimensional signal vector: = | | i =1,2,3,4 Elements of the signal vectors s i1 and s i2 are given as   𝑠 i (𝑡) Gray encoded i /p dibit QPSK signal phase (radians) Coordinates of message points s i1 s i2 11 + + 01 - + 00 - - 10 + - 𝑠 i (𝑡) Gray encoded i /p dibit QPSK signal phase (radians) Coordinates of message points s i1 s i2 11 01 00 10

31. QPSK: Signal space representation A QPSK signal has a two-dimensional signal constellation (i.e., N = 2) and four message points (i.e., M = 4) whose phase angles increase in a counterclockwise direction, as shown below. As with binary PSK, the QPSK signal has minimum average energy.

32. QPSK: Signal space representation

33. QPSK: Signal space representation

34. QPSK: Generator

35. QPSK: Generator The QPSK transmitter may be viewed as two BPSK generators that work in parallel, each at a bit rate equal to one-half the bit rate of the original binary sequence at the QPSK transmitter input. A distinguishing feature of the QPSK transmitter is the block labeled demultiplexer. The function of the demultiplexer is to divide the binary wave produced by the polar NRZ-level encoder into two separate binary waves, one of which represents the odd-numbered dibits in the incoming binary sequence and the other represents the even-numbered dibits in the incoming binary sequence

36. QPSK: Coherent Detector

37. QPSK: Coherent Detector The QPSK receiver is structured in the form of an in-phase path and a quadrature path, working in parallel as shown. The functional composition of the QPSK receiver is as follows: Pair of correlators : Supplied with a common input x(t). The two correlators are supplied with a pair of locally generated ONBF: synchronized with the transmitter. The correlator outputs, produced in response to the received signal x(t), are denoted by x 1 and x 2 , respectively.  

38. QPSK: Coherent Detector Pair of decision devices: Act on the correlator outputs x 1 and x 2 by comparing each one with a zero-threshold. Here, it is assumed that the symbols 1 and 0 in the original binary stream at the transmitter input are equally likely. If x 1 > 0, a decision is made in favor of symbol 1 for the in-phase channel output; on the other hand, if x 1 < 0, then a decision is made in favor of symbol 0. Similar binary decisions are made for the quadrature channel.

39. QPSK: Coherent Detector Multiplexer: the function of which is to combine the two binary sequences produced by the pair of decision devices. The resulting binary sequence so produced provides an estimate of the original binary stream at the transmitter input.

40. QPSK: Probability of error for coherent detection In a QPSK system operating on an AWGN channel, the received signal x(t) is defined by where w(t) is the sample function of a white Gaussian noise process of zero mean and power spectral density The two correlator outputs, x 1 and x 2 , are respectively defined as follows:  

41. QPSK: Probability of error for coherent detection The observable elements x 1 and x 2 are sample values of independent Gaussian random variables with mean values equal to + and + and with a common variance equal to The decision rule is now simply to say that: s 1 (t) was transmitted if the received signal point associated with the observation vector x falls inside region Z 1 ; s 2 (t) was transmitted if the received signal point falls inside region Z 2 , And so on for the other two regions Z 3 and Z 4 . An erroneous decision will be made if, for example, signal s 4 (t) is transmitted but the noise w(t) is such that the received signal point falls outside region Z 4 .  

42. QPSK: Probability of error for coherent detection The observable elements x 1 and x 2 are sample values of independent Gaussian random variables with mean values equal to + and + and with a common variance equal to