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Digital Signal Processing Module 1 Analysis of Discrete time Linear Time - Invariant Systems Objective: 1. To understand the representation of Discrete time signals 2. To analyze the causality and stability concepts of Linear Shift Invariant (LSI) systems Introduction: Digital signals are discrete in both time (the independent variable) and amplitude (the dependent variable). Signals that are discrete in time but continuous in amplitude are referred to as discrete-time signals. A discrete-time system is one that processes a discrete-time input sequence to produce a discrete-time output sequence. There are many different kinds of such systems. One of the important kinds is Linear Shift Invariant (LSI) systems. Description: Discrete time Signals and Sequences Discrete-time signals are data sequences. A sequence of data is denoted {x[n]} or simply x[n] when the meaning is clear. The elements of the sequence are called samples. The index n associated with each sample is an integer. If appropriate, the range of n will be specified. Quite often, we are interested in identifying the sample where n = 0. This is done by putting an arrow under that sample. For instance, 𝑥𝑛 = … ,0.35,1,1.5, −0.6, −2, … The arrow is often omitted if it is clear from the context which sample is x[0]. Sample values can either be real or complex. The terms “discrete-time signals” and “sequences” are used interchangeably. The time interval between samples is can be assumed to be normalized to 1 unit of time. So the corresponding normalized sampling frequency is 1Hz. If the actual sampling interval is T 1 seconds, then the sampling frequency is given by 𝑓𝑠 = 𝑇 Some Elementary Sequences i) Unit Impulse Sequence The unit impulse sequence is defined by This is depicted graphically in Figure 1.1. Note that while the continuous-time unit impulse function is a mathematical object that cannot be physically realized, the unit impulse sequence can easily be generated.

Figure 1.1: The Unit Impulse Sequence ii) Unit Step Sequence The unit step sequence is one that has amplitude of zero for negative indices and amplitude of one for non-negative indices. 0 𝑛<0 𝑢 𝑛 = 1 𝑛≥0 It is shown in Figure 1.2. Figure 1.2: The Unit Step Sequence iii) Exponential sequences The general form is If A and α are real numbers then the sequence is real. If 0 < α < 1 and A is positive, then the sequence values are positive and decrease with increasing n:

iv) Sinusoidal Sequences A sinusoidal sequence has the form This function can also be decomposed into its in-phase xi[q] and quadrature xq[n] components. This is a common practice in communications signal processing. It is shown in Figure 1.3. Figure 1.3: The Sinusoidal Sequence v) Complex Exponential Sequences Complex exponential sequences are essentially complex sinusoids. vi) Random Sequences The sample values of a random sequence are randomly drawn from a certain probability distribution. They are also called stochastic sequences. The two most common distributions are the Gaussian (normal) distribution and the uniform distribution. The zero-mean Gaussian distribution is often used to model noise. Figure 1.4 and Figure 1.5 show examples of uniformly distributed and Gaussian distributed random sequences respectively.

Figure 1.4: Uniformly distributed random sequence with amplitudes between -0.5 and 0.5. Figure 1.5: Gaussian distributed random sequence with zero mean and unit variance. Types of Sequences The discrete-time signals that we encounter can be classified in a number of ways. Some basic classifications that are of interest to us are described below. i) ii) iii) Real vs. Complex Signals A sequence is considered complex at least one sample is complex-valued. Finite vs. Infinite Length Signals Finite length sequences are defined only for a range of indices, say N1 toN2. The length of this finite length sequence is given by |N2- N1 + 1|. Causal Signals A sequence x[n] is a causal sequence if x[n] = 0 for n < 0.

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