Advanced Digital Systems Lecture 1 PDF

Advanced Digital Systems Dr. Sahar Hamed Outline Introduction Analog to Digital Conversion Nyquist–Shannon Sampling Theorem Continuous Time and Discrete Time Signals  Introduction Continuous-Time Signal X(t) is defined for all time t contained in some interval on the...

Advanced Digital Systems Dr. Sahar Hamed Outline Introduction Analog to Digital Conversion Nyquist–Shannon Sampling Theorem Continuous Time and Discrete Time Signals  Introduction Continuous-Time Signal X(t) is defined for all time t contained in some interval on the real line. Discrete-Time Signal X[n] a sequence of values that correspond to particular instants in time. Examples of Continuous & Discrete-Time Signals Most of the signals are analog in nature: Voice Video Transducer/Sensor output Examples of Discrete-Time Signals: Average budget Crime rate Total population Analog Signals Vs Digital Signals Analog Digital Analog signals are of much higher Digital signal processing is more secure density and can present more accurate because digital information can be easily information. encrypted and compressed. Analog signals provide a more accurate Digital signals can be transmitted over representation of changes in physical long distances. phenomena, such as sound, light, temperature, position, or pressure. Analog signals are subject to noise and Digital systems and processing are distortion. typically more complex and consume higher power dissipation. Analog to Digital Conversion Analog Analog input output Signal x(t) Signal y(t) A/D Digital D/A converter Signal Processor converter Digital Signal Processing Analog to Digital Conversion A/D conversion can be viewed as a three-step process 7 Analog to Digital Conversion Sampling: Convert the continuous time & value signal to discrete-time & continuous value signal. Quantization: Convert the discrete-time & continuous value signal to discrete-time & value signal Coding: Convert the discrete-time & value signal to a digital data format 8 Different sampling rate 9 Nyquist–Shannon Sampling Theorem If an analog signal is sampled at a rate that exceeds the signal’s highest frequency by at least a factor of two, the original analog signal can be perfectly recovered from the discrete values produced by sampling. 𝑓𝑠 ≥ 2𝑓𝑚𝑎𝑥 10 The sampling rate The sampling rate If a signal is sampled at T = 1mS, the sampling rate will be fs = 1/1m = 1KHZ (i.e., 1000 samples per second) 11 Example What is the sampling rate for a voice signal that has frequencies up to 3KHZ? Fs = 2 * 3000 = 6000samples/sec (HZ) What is the sampling rate for a sound that has frequencies up to 30KHZ? Fs = 2 * 30000 = 60000samples/sec (HZ) 12 Test your understanding Determine the Nyquist sampling rate of the signal. 1. X(t) = 3 Sin (5000π + 17) 2. S(t) = 3Cos (50πt) + 10 Sin (300πt) – Cos (100πt) 13 Signal Representation Continuous time signal Signal Representation Discrete time signal X[n] Discrete Time Signal – Time Shifting Discrete Time Signal – Reflection Discrete Time Signal – Time Scaling Exponential Signals 𝑥 𝑛 = 𝐴 ∝𝑛 Test Your Understanding Plot x[n] & determine y[n] Periodic Signals For a continuous time signal: The signal is said to be periodic if x(t) = x(t+T) Periodic Signals For a discrete time signal: The signal is said to be periodic if x[n] = x[n+N] Even & Odd Signal X(-t) = X(t) X[-n]=X[n] X(-t) = -X(t) X[-n]= -X[n] Even & Odd decomposition of discrete time signals Any signal can be broken into a sum of two signals: Odd & Even Discrete-Time Unit Impulse Discrete-Time Unit Step Relation between Unit Impulse & Unit Step Reading List Signals & Systems, Allan Oppenheim & Allan Wilsky Chapter 1 Thank You