COSC 4214: Digital Communications Course Instructor: Amir Asif Contact Information: Office: CSB 3028 asif@cse.yorku.ca (416) 736-2100 X70128 URL: http://www.cs.yorku.ca/course/4214 Text: B. Sklar, Digital Communications: Fundamentals and Applications, NJ: Prentice Hall, 2nd edition, 2001, ISBN # 0-13-084788-7. Class Schedule: TR 10:00 - 11:30  (CB 120) Assessment: Assignment / Quiz: 20% Projects: 15% Mid-term: 25% Final: 40% Office Hours: Instructor: CSB 3028, TR 12:00 - 13:00

Course Overview

Transformations in Digital Communications

Objectives of COSC4214 Introduce a mathematical representation for noise Process a random process such as noise with a linear system Learn about baseband communications: single signal transmission at low frequencies (Baseband Modulation) Learn about bandpass or broadband communications: multiple signal transmission at high frequencies (Bandpass Modulation) Detect signal in the presence of noise (Matched filtering) How to control error using error detection and correction techniques (Channel coding) Learn advance topics like Spread Spectrum techniques Demystify terminology !!!!!!

Why Digital? Digital signals are less subject to distortion and interference than analog signals. Digital signal processing is more reliable and less expensive than analog signal processing Digital communication techniques lend themselves naturally to channel coding for: Protection against jamming and interference Secure transmission to prevent eavesdropping

Classification of Signals (1) Deterministic versus Random Signals. Continuous-time (CT) versus Discrete-time (DT) Signals. Analog versus Digital Signals. Periodic versus Aperiodic Signals. Energy versus Power Signals. Odd and Even Signals.

Classification of Signals (2): Deterministic vs Random Signals Deterministic Signals: Defined for all time No uncertainty with respect to the value of the signal Represented using a mathematical expression, e.g., x(t) = sin(5pt + 30o). Random Signals: Are not known accurately for all instants of times Different observations may lead to different results Statistical properties such as mean, variance, or probability density function (pdf) are used to define the random signal

Classification of Signals (3): Continuous-time vs Discrete-time Signals Continuous-time Signals: Defined for all instants of time. Discrete-time Signals: Defined at discrete values of time. Activity 1: Plot the CT signal x(t) = sin(5pt + 30o). Discretize the CT signal with an uniform sampling period of Ts = 0.25s. Sketch the resulting waveform.

Classification of Signals (4): Analog vs Digital Signals Analog Signals: Defined for all instants of time. Amplitude can take on any value. Digital Signals: Defined at discrete values of time. Amplitude is restricted to finite set of values. t

Classification of Signals (5): Analog vs Digital Signals Analog Signals: Defined for all instants of time. Amplitude can take on any value. Digital Signals: Defined at discrete values of time. Amplitude is restricted to finite set of values. CT Signal DT Signal Sampling k

Classification of Signals (6): Analog vs Digital Signals Analog Signals: Defined for all instants of time. Amplitude can take on any value. Digital Signals: Defined at discrete values of time. Amplitude is restricted to finite set of values. CT Signal DT Signal Sampling Digital Signal Quan-tization k

Classification of Signals (7): Periodic vs Aperiodic Signals Periodic signals: A periodic signal x(t) is a function of time that satisfies the condition x(t) = x(t + T0) for all t where T0 is a positive constant number and is referred to as the fundamental period of the signal. Fundamental frequency (f0) is the inverse of the period of the signal. It is measured in Hertz (Hz =1/s). Nonperiodic (Aperiodic) signals: are those that do not repeat themselves. Activity 2: For the sinusoidal signals (a) x[k] = sin (5pk) (b) y[k] = sin(k/3) determine the fundamental period K0 of the DT signals.

Classification of Signals (8): Even vs Odd Signals Even Signal: A CT signal x(t) is said to be an even signal if it satisfies the condition x(-t) = x(t) for all t. Odd Signal: The CT signal x(t) is said to be an odd signal if it satisfies the condition x(-t) = -x(t) for all t. Even signals are symmetric about the vertical axis or time origin. Odd signals are antisymmetric (or asymmetric) about the time origin.

Classification of Signals (9): Even vs Odd Signals Signals that satisfy neither the even property nor the odd property can be divided into even and odd components based on the following equations: Even component of x(t) = 1/2 [ x(t) + x(-t) ] Odd component of x(t) = 1/2 [ x(t) - x(-t) ] Activity 3: For the signal do the following: (a) sketch the signal (b) evaluate the odd part of the signal (c) evaluate the even part of the signal.

Classification of Signals (10): Energy vs Power Signals Activity 4: a. Consider the sinusoidal signal x(t) = cos(0.5pt). Choosing a time period T0 = 4, determine the average power of x(t). b. Consider the signal x(t) that equals 5 cos(pt) for the interval -1 <= t <= 1 and is 0 elsewhere. Calculate the energy of x(t). c. Calculate the energy of the signal x[k] = (0.5)k for k >= 0 and is 0 elsewhere.

Power vs. Energy Signals (2) Energy Signals: have finite total energy for the entire duration of the signal. As a consequence, total power in an energy signal is 0. Power Signals: have non-zero power over the entire duration of the signal. As a consequence, the total energy in a power signal is infinite. Periodic signals are always power signals with power given by

Elementary Signal: Unit Impulse Function Impulse function is defined from the following properties: Activity 5: Solve the integral

Deterministic Signals: Energy and Spectral Density An energy (aperiodic) signal x(t) can be represented by its Fourier transform X(f) The energy spectral density (ESD) of an energy signal is defined as A power (periodic) signal x(t) can be represented by its Fourier Series The power spectral density (PSD) of a power signal is defined as Activity 6: Calculate the ESD or PSD of the following functions: (i) x(t) = d(t); (ii) x(t) = cos(2pf0t). From the ESD or PSD, determine the energy and power of each signal.

Deterministic Signals: Autocorrelation The autocorrelation of an energy (aperiodic) signal x(t) is defined as The autocorrelation satisfies the following properties Activity 7: Calculate the autocorrelation and power spectral density for the gate function x(t) = 1 for -0.5 ≤ t ≤ 0.5; 0 elsewhere.