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Signals and Systems Defined

A signal is any physical phenomenon which conveys information
Systems respond to signals and produce new signals
Excitation signals are applied at system inputs and response signals are produced at system outputs

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A Communication System as a System Example

A communication system has an information signal + noise signals
This is an example of a system that consists of an interconnection of smaller systems

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Signal Types

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Conversions Between Signal Types

Sampling

Quantizing

Encoding

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Message Encoded in ASCII

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Noisy Message Encoded in ASCII

Progressively

noisier

signals

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Bit Recovery in a Digital Signal Using Filtering

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Image Filtering to Aid Perception

Original X-Ray Image

Filtered X-Ray Image

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Discrete-Time Systems

In a discrete-time system events occur at points in time but not

between those points. The most important example is a digital

computer. Significant events occur at the end of each clock

cycle and nothing of significance (to the computer user) happens

between those points in time.

Discrete-time systems can be described by difference (not

differential) equations. Let a discrete-time system generate an

excitation signal y[n] where n is the number of discrete-time

intervals that have elapsed since some beginning time n = 0.

Then, for example a simple discrete-time system might be

described by

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Discrete-Time Systems

The equation

says in words

“The signal value at any time n is 1.97 times the signal value at the

previous time [n -1] minus the signal value at the time before that

[n - 2].”

If we know the signal value at any two times, we can compute its

value at all other (discrete) times. This is quite similar to a

second-order differential equation for which knowledge of two

independent initial conditions allows us to find the solution for all

time and the solution methods are very similar.

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Discrete-Time Systems

We could solve this equation by iteration using a computer.

yn = 1 ; yn1 = 0 ;

while 1,

yn2 = yn1 ; yn1 = yn ; yn = 1.97*yn1 - yn2 ;

end

We could also describe the system

with a block diagram.

Initial Conditions

(“D” means delay one unit in discrete

time.)

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Discrete-Time Systems

With the initial conditions y[1] = 1 and y[0] = 0 the response

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Feedback Systems

In a feedback system the response of the system is “fed back”

and combined with the excitation is such a way as to optimize

the response in some desired sense. Examples of feedback

systems are

Temperature control in a house using a thermostat
Water level control in the tank of a flush toilet.
Pouring a glass of lemonade to the top of the glass without

overflowing.

A refrigerator ice maker that keeps the bin full of ice

but does not make extra ice.

Driving a car.

Feedback systems can be continuous-time or discrete-time

or a mixture of the two.

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Feedback Systems

Below is an example of a discrete-time feedback system. The

response y[n] is fed back through two delays and gains b and c

and combined with the excitation x[n]. Different values of a,

b and c can create dramatically different responses to the same

excitation.

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Feedback Systems

Responses to an excitation that changes from 0 to 1 at n = 0.

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Sound Recording System

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Recorded Sound as a Signal Example

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