1. Introduction to Signals and Systems
Chapter 1

2. 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
11/16/2018
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

3. A Communication System as a System Example
A communication system has an information signal plus noise signals
This is an example of a system that consists of an interconnection of smaller systems
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M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

4. M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
Signal Types
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M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

5. Conversions Between Signal Types
Sampling
Quantizing
Encoding
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6. Message Encoded in ASCII
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7. Noisy Message Encoded in ASCII
Progressively
noisier
signals
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M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

8. Bit Recovery in a Digital Signal Using Filtering
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M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

9. The Four Fourier Methods
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10. CT Fourier Series Definition
The Fourier series representation, , of a signal, x(t),
over a time, , is
where X[k] is the harmonic function, k is the harmonic
number and (pp ). The harmonic function
can be found from the signal as
The signal and its harmonic function form a Fourier series
pair indicated by the notation,
11/16/2018
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

11. The Trigonometric CTFS
The fact that, for a real-valued function, x(t),
also leads to the definition of an alternate form of the CTFS,
the so called trigonometric form.
where
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M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

12. M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl
Definition of the CTFT
Forward
Inverse
f form
w form
Forward
Inverse
Commonly-used notation: or
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M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

13. Relations Among Fourier Methods
Discrete Frequency
Continuous Frequency CT DT
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M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

14. Generalization of the CTFT: Laplace Transform
The CTFT expresses a time-domain signal as a linear combination of complex sinusoids of the form, In the generalization of the CTFT to the Laplace transform the complex sinusoids become complex exponentials of the form, ,where s can have any complex value . Replacing the complex sinusoids with complex exponentials leads to this definition of the Laplace transform,
A function and its Laplace transform form a transform pair which is conveniently indicated by the notation,
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M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

15. Relation to the Laplace Transform
The z transform is to DT signals and systems what the Laplace transform is to CT signals and systems
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M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

16. Definition: z-Transform
The z transform can be viewed as a generalization of the
DTFT or as natural result of exciting a discrete-time LTI system with its eigenfunction. The DTFT is defined by
If a strict analogy with the Laplace transform were made W would replace w, S would replace s, S would replace s, a summation would replace the integral and the z transform would be defined by
11/16/2018
M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl