Introduction to Signals and Systems

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 • Excitationsignals are applied at systeminputs and responsesignals are produced at systemoutputs 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 M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

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

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

6. Message Encoded in ASCII M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

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

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

9. The Four Fourier Methods M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

10. 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, . CT Fourier Series Definition 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 M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

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

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