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����PRESENTATION ON �DISCRETE TIME SIGNAL

  • BRANCH-E & TC ENGG
  • SUBJECT- DIGITAL SIGNAL PROCESSING
  • CHAPTER – 2 – DISCRETE TIME SIGNAL & SYSTEM
  • TOPIC- DISCRETE TIME SIGNAL
  • SEM-6TH
  • FACULTY – Er. ARADHANA DAS (Sr. LECTURER E & TC ENGG DEPARTMENT)
  • AY-2021-2022, SUMMER-2022

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2.0 Introduction

  • Signal: something conveys information, represented mathematically as functions of one or more independent variables. Classified as:
  • Continuous-time (analog) signals, discrete-time signals, digital signals
  • Signal-processing systems are classified along the same lines as signals: Continuous-time (analog) systems, discrete-time systems, digital systems

2

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2.2 Discrete-Time System

  • Discrete-Time System is a trasformation or operator that maps input sequence x[n] into an output sequence y[n].
  • y[n]=T{x[n]};

x[n], y[n]: discrete-time signal

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T{‧}

x[n]

y[n]

Discrete-Time System

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EX. 2.2 The Ideal Delay System

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  • If is a positive integer: the delay of the system, Shift the input sequence to the right by samples to form the output .
  • If is a negative integer: the system will shift the input to the left by samples, corresponding to a time advance.

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Properties of Discrete-time systems�2.2.1 Memoryless (memory) system

  • Memoryless systems:

the output y[n] at every value of n depends only on the input x[n] at the same value of n

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Example 2.4 A Memoryless System

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Properties of Discrete-time systems�2.2.2 Linear Systems

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  • If

T{‧}

T{‧}

T{‧}

T{‧}

T{‧}

additivity property

homogeneity or scaling

同(齐)次性 property

  • principle of superposition
  • and only If:

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Example of Linear System

  • Ex. 2.5 Accumulator system

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for arbitrary

when

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Example 2.6 Nonlinear Systems

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  • Method: find one counterexample
  • counterexample
  • For
  • For
  • counterexample

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Properties of Discrete-time systems�2.2.3 Time-Invariant Systems

  • Shift-Invariant Systems

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T{‧}

T{‧}

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Example of Time-Invariant System

  • Ex. 2.7 The Accumulator as a Time-Invariant System

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Properties of Discrete-time systems� 2.2.4 Causality

  • A system is causal if, for every choice of , the output sequence value at the index depends only on the input sequence value for .

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Ex. 2.9 The Forward and Backward Difference Systems

  • Forward difference system is not Causal

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  • Backward difference system is Causal

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Properties of Discrete-time systems� 2.2.5 Stability

  • Bounded-Input Bounded-Output (BIBO) Stability: every bounded input sequence produces a bounded output sequence.

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if

then

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Ex. 2.10 Testing for Stability or Instability

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if

then

is stable

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  • Accumulator system

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  • Accumulator system is not stable

Ex. 2.10 Testing for Stability or Instability

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2.3 Linear Time-Invariant (LTI) Systems

  • Impulse response

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T{‧}

T{‧}

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Finite-duration impulse response (FIR) systems

  • The impulse response of the system has only a finite number of nonzero samples.
  • The FIR systems always are stable.

such as:

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Infinite-duration impulse response (IIR)

  • The impulse response of the system is infinite in duration

Stable IIR System:

Unstable system

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Finite-duration impulse response (FIR) systems

  • The impulse response of the system has only a finite number of nonzero samples.
  • The FIR systems always are stable.

such as:

Review

Moving Average systems

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Infinite-duration impulse response (IIR)

  • The impulse response of the system is infinite in duration

Stable IIR System:

Unstable system

Review

Accumulator system:

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