D R . T A R E K T U T U N J I

P H I L A D E L P H I A U N I V E R S I T Y , J O R D A N

2 0 1 4

Discrete Time Signals and Systems

Discrete Time Signals

Introduction

The basic theory of discrete-time signals and systems is similar to continuous-time signals and systems. However, there are some differences:

Discrete-time signals result from sampling of continuous-time signals and are only available at uniform times determined by the sampling period

Discrete-time signals depend on an integer variable n

The radian discrete frequency cannot be measured and depends on the sampling period

Introduction

Discrete-time periodic signals must have integer periods.

This imposes some restrictions for example it is possible to have discrete-time sinusoids that are not periodic, even if they resulted from the uniform sampling of continuous-time sinusoids.

Basic math operations:

Integrals are replaced by sums

Derivatives are replaced by finite differences

Differential equations are replaced by difference equations

Discrete Time Signals

A sampled signal x(nTs) = x(t)|t=nTs is a discrete-time signal x[n] that is a function of n only.

Once the value of Ts is known, the sampled signal only depends on n, the sample index.

Nyquist sampling rate condition

Example

Periodic Signals

Periodic Signals

Example

Periodic Discrete Sinusoid

Finite Energy and Finite Power

Example

Time Shift

Time Shift

Time Reflection

Odd and Even Signals

Discrete Time Exponential Signal

Discrete Time Exponentials

Discrete Frequency

Discrete Time Sinusoidal Signals

Unit Step and Impulse

Examples

Discrete Time Systems

Difference Equations

Recursive and Non-recursive Systems

Properties

Linearity

Time invariance

Stability

Causality

Linearity and Time-invariant

Example

Example

Causality and Stability

Convolution

Example

Conclusion

Discrete-Time signals are the result of sampling continuous-time signals

Discrete-time signals have discrete radian frequency that

varies between -p and p

Discrete-Time systems properties are similar to continuous-time systems: Linear, time-invariant, causal, and stable

Convolution operation is used in the time-domain Auto Regressive Moving Average (ARMA) represent a class of

linear discrete-time systems

References

Chapter 8. Signals and Systems using MATLAB by Luis Chaparro. Elsevier Publisher 2011.