CE 331: SIGNALS & SYSTEMS

Topic 1

Introduction to Signals

(Oppenheim et. al. pp 1-38)

These slides are based on the lecture notes used in 6.003 at MIT which are co-authored by Qing Hu, D. Boning,

D. Freeman, T. Weiss, J. White, and A. Willsky.

German Jordanian University

Spring 2012/2013

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Signal and Systems

• Signals are functions of one or more independent

variables to tell information about certain

phenomena

• Systems process input signals to produce output

signals

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Examples of Signals

• Electrical signals --- voltages and currents in a

circuit

• Acoustic signals --- audio or speech signals

(analog or digital)

• Video signals --- intensity variations in an image

(e.g. a CAT scan)

• Biological signals --- sequence of bases in a gene

•

• In CE 331, we will treat noise as unwanted signals

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Signal Classification

Type of Independent Variable:

• Time is often the independent variable. Example:

the electrical activity of the heart recorded with

chest electrodes –– the electrocardiogram (ECG or

EKG)

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Signal Classification

Type of Independent Variable:

• The independent variables can also be spatial.

Example: the MRI image.

In this example, the signal is

the intensity as a function of

the spatial variables x and y.

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Signal Classification

Independent Variable Dimensionality:

• An independent variable can be 1-D (t in the EKG) or 2-D (x, y in an image).

• In this course, we will focus on 1-D for mathematical

simplicity but the results can be extended to 2-D or even

higher dimensions. Also, we will use a generic time t for

the independent variable, whether it is time or space.

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Continuous-time (CT) Signals

• Most of the signals in the physical world are CT signals, since the time scale is infinitesimally fine, so are the

spatial scales.

• Example: voltage & current, pressure, temperature,

velocity, etc.

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Discrete-time (DT) Signals

• x[n], n — integer, time varies discretely

• Examples of DT signals in nature:

– DNA base sequence

– Population of the nth generation of certain species

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Notation

• In this course:

x(t) is used to denote continuous-time (CT) signal

x[n] is used to denote discrete-time (DT) signal

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Many human-made Signals are DT

Why DT? — Can be processed by modern digital

computers and digital signal processors (DSPs).

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Signals with symmetry

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Signals with symmetry

The oddness

of x(t) or

x[n] implies

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Signals with symmetry

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Right- and Left-Sided Signals

• A right-sided signal is zero for t < T and a leftsided signal is zero for t > T, where T can be positive or negative.

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Bounded and Unbounded Signals

• Whether the output signal of a system is bounded or

unbounded determines the stability of the system.

• As time tends to infinity, the absolute value of the signal magnitude can either:

1. Continuously decrease and/or increase (or stay

constant) but remain within a bounded range

2. Continuously increase to very large values without any

bound

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Bounded and Unbounded Signals

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Bounded and Unbounded Signals

• Examples of bounded signals:

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Bounded and Unbounded Signals

• Examples of Unbounded signals:

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Real and Complex Signals

• If x is a complex quantity, then it has:

– A real and imaginary part, or equivalently

– a magnitude and a phase angle

We will use whichever form that is convenient.

• A very important class of signals is complex

exponentials:

– CT signals of the form x(t) = est

– DT signals of the form x[n] = zn

where z and s are complex numbers.

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Real and Complex Signals

• If x is a complex quantity, then it has:

– A real and imaginary part, or equivalently

– a magnitude and a phase angle

We will use whichever form...