2. Discrete Time Signals_2019(1).pdf - Outline Discrete-Time Signals Dr Bokamoso Basutli Lecturer and Group Leader(SPNS Department of Electrical

2. Discrete Time Signals_2019(1).pdf - Outline...

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Outline Discrete-Time Signals Dr. Bokamoso Basutli Lecturer and Group Leader (SPNS) Department of Electrical, Computer and Telecommunications Engineering Digital Signal Processing Lecture 2: Discrete-Time Signals and Systems
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Outline Discrete-Time Signals Outline Discrete-Time Signals
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Outline Discrete-Time Signals Elementary Discrete-Time Signals (1) Unit sample sequence (a.k.a. Kronecker delta function): δ [ n ] = 0 , n 6 = 0, 1 , n = 0. (2) Unit Step Signal: u [ n ] = 1 , n 0, 0 , n < 0. (3) Unit Ramp Signal: u r [ n ] = n , n 0, 0 , n < 0. NOTE: δ [ n ] = u [ n ] - u [ n - 1] = u r [ n + 1] - 2 u r [ n ] + u r [ n - 1] = u r [ n + 1] - u r [ n ]
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Outline Discrete-Time Signals Unit Step Manipulations A unit step sequence, u [ n ] is given as; u [ n ] = 1 n 0, 0 n < 0. or in terms of δ [ n ] u [ n ] = n X k = -∞ δ [ k ] = X k =0 δ [ n - k ] which is analogous to the continous-time unit step being written as Figure 1: Rectangular pulses using unit step sequence. u ( t ) = Z t -∞ δ ( λ ) d λ = Z 0 δ ( t - λ ) d λ
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Outline Discrete-Time Signals Signal Symmetry Even Signal: x [ - n ] = x [ n ] Odd Signal: x [ - n ] = - x [ n ]
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Outline Discrete-Time Signals Signal Symmetry Even Signal component: x e [ n ] = 1 2 [ x [ n ] + x [ - n ]] Odd Signal component: x o [ n ] = 1 2 [ x [ n ] + - x [ - n ]] Note: x [ n ] = x e [ n ] + x o [ n ]
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Outline Discrete-Time Signals Simple Manipulation of Discrete-Time Signals Transformation of independent variable: time shift: n n - k , k Z . What if k / Z time scale: n α n , α Z . What if α / Z Additional, multiplication and scaling: amplitude scaling: y [ n ] = Ax [ n ] , < n < sum: y [ n ] = x 1 [ n ] + x 2 [ n ] , < n < product: y [ n ] = x 1 [ n ] x 2 [ n ] , < n <
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Outline Discrete-Time Signals Simple Manipulation of Discrete-Time Signals Sequence shift or delay is obtained by letting n n - k which defines a new signal y [ n ] = x [ n - k ] . If k > 0 then y [ n ] is delayed and if k < 0 then y [ n ] is advanced, with respect to x [ n ]. Unit sample sequence or impulse, δ [ n ] is the discrete-time equivalent to δ [ n ] = 0 n 6 = 0, 1 n = 0. Any sequence can be written as a linear combintaion of δ [ n ], e.g. x [ n ] = X k = -∞ x [ k ] δ [ n - k ]
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Outline Discrete-Time Signals Tutorial: Simple Manipulation of Discrete-Time Signals Consider the sequence x [ n ] given in the figure below. Determine x [ n ] = x [ n + 1]
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Outline Discrete-Time Signals Tutorial: Simple Manipulation of Discrete-Time Signals Determine x [ 3 2 n + 1]
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Outline Discrete-Time Signals Tutorial: Simple Manipulation of Discrete-Time Signals Graph of x [ 3 2 n + 1]
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Outline Discrete-Time Signals Input-Output Description Input-output description (exact structure of system is unknown or ignored): y [ n ] = T { x [ n ] } “black box” representation: x [ n ] T -→ y [ n ]
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Outline Discrete-Time Signals Discrete-Time Systems Discrete-Time System A discrete-time system is an operator that maps an input sequence into an output sequence. y [ n ] = T { x [ n ] } Note that T {·} is an operator that maps the input sequence to the output sequence. Figure 2: Discrete-time system block diagram definition.
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Outline Discrete-Time Signals Classifiying Discrete-Time Systems Discrete-time system can be classified based on their properties, such as: static or dynamic ( memory or memoryless,) linearity or non-linearity, time-invariant or time-varying, causal or non-causal, bounded-input bounded-output (BIBO stability) or unbounded.
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