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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.
