Introduction to Signals and Systems Chapter 1

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

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

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

Conversions Between Signal Types Sampling Quantizing Encoding 11/16/2018 M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

Message Encoded in ASCII 11/16/2018 M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

Noisy Message Encoded in ASCII Progressively noisier signals 11/16/2018 M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

Bit Recovery in a Digital Signal Using Filtering 11/16/2018 M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

The Four Fourier Methods 11/16/2018 M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

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. 212-215). 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

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

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

Relations Among Fourier Methods Discrete Frequency Continuous Frequency CT DT 11/16/2018 M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

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

Relation to the Laplace Transform The z transform is to DT signals and systems what the Laplace transform is to CT signals and systems 11/16/2018 M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

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