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1) Administrative details

2) Signals

“Figures and images used in these lecture notes by permission,copyright 1997 by Alan V. Oppenheim and Alan S. Willsky”

Signals and SystemsSpring 2004Lecture #1 (2/3/04)

Prof. Qing Hu

(Slides thanks to D. Boning, T. Weiss,J. White, and A. Willsky)

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Signals and Systems

6.003 is about using mathematical techniques to helpanalyze and synthesize systems which process signals.

• Signals are variables that carry information.

• Systems process input signals to produce outputsignals.

Today: Signals, next lecture: Systems.

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Examples of signals

• Electrical signals --- voltages and currents in a circuit

• Acoustic signals --- audio or speech signals (analog ordigital)

• Video signals --- intensity variations in an image (e.g. aCAT scan)

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

• In 6.003, we will treat noise as unwanted signals

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Signal ClassificationType of Independent Variable

Time is often the independent variable. Example: theelectrical activity of the heart recorded with chestelectrodes –– the electrocardiogram (ECG or EKG).

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The variables can also be spatial

Eg. Cervical MRI

In this example, thesignal is the intensity asa function of the spatialvariables x and y.

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Independent Variable Dimensionality

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

In 6.003, focus on 1-D for mathematical simplicity but the results can beextended to 2-D or even higher dimensions. Also, we will use a generictime 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 thespatial scales. E.g. voltage & current, pressure,temperature, velocity, etc.

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Discrete-time (DT) Signals

• Examples of DT signals in nature:— DNA base sequence— Population of the nth generation of certain species

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• Notation in 6.003: x(t) –– CT, x[n] –– DT

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

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Many human-made Signals are DT

Ex.#1 Weekly Dow-Jones

industrial average

Why DT? — Can be processed by modern digital computersand digital signal processors (DSPs).

Ex.#2 digital image

Ex#3. ticking clock M

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Mandrill ExampleUnblurred Image & No Noise

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Mandrill ExampleBlurred Image (bad focus)

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Mandrill ExampleUnblurred Image – 0.1% Noise (too dark)

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Signals with symmetry

• Periodic signalsCT x(t) = x(t + T)

DT x[n] = x[n + N]

• Ex. 60-Hz power line, computer clock, etc.

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Signals with symmetry (continued)

• Even and odd signalsEven

Odd

x(t) = x(−t) or x[n] = x[−n]

x(t) = − x(−t) or x[n] = − x[−n]

x(0) = 0or x[0] = 0

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Signals with symmetry (continued)

• Any signals can be expressed as a sum of Even andOdd signals. That is:

x(t) = xeven(t) + xodd(t) ,

where

xeven(t) = [x(t) + x(-t)]/2 , xodd(t) = [x(t) - x(-t)]/2 .

Ex. DT unit-stepfunction.

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Right- and Left-Sided Signals

A right-sided signal is zero for t < T and a left-sided signalis 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 orunbounded determines the stability of the system.Eg. Demo, the inverted pendulum, unstable but can bestabilized using a feedback control system.

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Real and Complex SignalsA very important class of signals is:

• CT signals of the form x(t) = est

• DT signals of the form x[n] = zn

where z and s are complex numbers. For both exponentialCT and DT signals, x is a complex quantity and has:

• a real and imaginary part, or equivalently

• a magnitude and a phase angle.

We will use whichever form that is convenient.

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For example, suppose s = j!/8 and z = e j!/8, then thereal parts are

R {x(t) = est} = R {e j!t/8} = cos(!t/8),

R {x[n] = zn} = R {e j!n/8} = cos[!n/8].

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Summary• We are awash in a sea of signal –– sounds, visual,

electrical, thermal, mechanical, etc.

• Signals can be time-varying or spatially varying, canbe a function of multiple variables. However, in6.003, we will mostly deal with 1D signals and use ageneric time t to represent the variable,whether it istime, space, or something else.

• DT signals and systems become more and moreimportant for signal processing, and will be a majorpart in 6.003.