Signals and Systems

Topics: Real number, Time, Complex number Pages: 7 (769 words) Published: April 14, 2013
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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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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...