111 Lecture 2 Signals and Systems (II) Principles of Communications Fall 2008 NCTU EE Tzu-Hsien Sang

2 Outlines Signal Models & Classifications Signal Space & Orthogonal Basis Fourier Series &Transform Power Spectral Density & Correlation Signals & Linear Systems Sampling Theory DFT & FFT 2

More on LTI Systems A system is BIBO if output is bounded, given any bounded input. A system is causal if: current output does not depend on future input; or current input does not contribute to the output in the past. 3

Paley-Wiener Condition: Remarks: (1) |H(f)| cannot grow too fast. (2) |H(f)| cannot be exactly zero over a finite band of frequency. 2 nd ver.: 4

Eigenfunctions of LTI Systems Another way of taking complicated things part. Instead of trying to find a set of orthogonal basis functions, let’s look for signals that will not be changed “fundamentally” when passing them through an LTI system. Why? Consider the key words: analysis/synthesis. Note: Eigen-analysis is not necessarily consistent with orthogonal basis analysis. 5

If, where  is a constant, then  is the eigenvalue for the eigenfunction g(t). Let 6

7 (Cross)correlation functions related by LTI systems: Note: In proving them, we use: Filters!!!

8 Since almost any input x(t) can be represented by a linear combination of orthogonal sinusoidal basis functions, we only need to input to the system to characterize the system’s properties, and the eigenvalue carries all the system information responding to. (Frequency response!!!) In communications, signal distortion is of primary concern in high-quality transmission of data. Hence, the transmission channel is the key investigation target.

9 Three major types of distortion caused by a transmission channel: 1. Amplitude distortion: linear system but the amplitude response is not constant. 2. Phase (delay) distortion: linear system but the phase shift is not a linear function of frequency. (Q: What good is linear phase?) 3. Nonlinear distortion: nonlinear system

10 Example: Group Delay

11

12 Example: Ideal general filters

13 Realizable filters approximating ideal filters

14 The Uncertainty Principle It can be argued that a narrow time signal has a wide (frequency) bandwidth, and vice versa:

15 Sampling Theory You’ve probably heard of “signal processing.” But how to process a signal? For instance, the rectifier – max{x(t), 0}. But, how to do Fourier transform of an arbitrary signal x(t)? Computers seem a good idea. But computers can only work on numbers. We need to “transform” the signal first into numbers. Q: Tell discrete signals from digital signals.

16 Ideal sampling signal: impulse train (an analog signal), T: the sampling period Analog (continuous-time) signal: Sampled (continuous-time) signal: Hopefully the math becomes easier in ideal case. The concept actually is harder.

17 Aliasing: If The replicas of X(f) overlap in frequency domain. That is, the higher frequency components of overlap with the lower frequency components of X(f-f s ).

18 –Nyquist Sampling Theorem: Let x(t) be a bandlimited signal with X(f) = 0 for. (i.e., no components at frequencies greater than W.) Then x(t) is uniquely determined by its samples if.

19 In other words, oversampling preserves all the information that x(t) contains. It is possible to reconstruct x(t) purely by its samples. Ideal reconstruction filter (interpretation in frequency domain: In time domain:

20 Two types of reconstruction errors

21 DFT & FFT You can view DFT as a totally new definition for a totally different set of signals. Or you can try to connect it to the Fourier Transform.

22

Fast Fourier Transform (FFT) is not a new transform, it is simply a fast way to compute DFT. So, don’t use FFT to denote the object that you want to compute; only use it to denote the tool that you use to compute it. (Gauss knew the method already!) Application example: Fast convolution via FFT: 23