1. EE313 Linear Systems and Signals Fall 2010 Initial conversion of content to PowerPoint by Dr. Wade C. Schwartzkopf Prof. Brian L. Evans Dept. of Electrical and Computer Engineering The University of Texas at Austin Signal Energy

2. 15 - 2 Time/Frequency Domains Signal Time domain: waveform, signal duration, waveform decay, causality, periodicity Frequency domain: sinusoidal components, relative amplitudes and phases LTI system Impulse response: width indicates system time constant, a.k.a. response time, and amount of dispersion (spreading) Frequency response: bandwidth filter selectively transmits certain frequency components and suppresses the other, transfer functions, stability

3. 15 - 3 Time/Frequency Domains Time-Frequency Fourier transform gives a global average of the frequency content of a signal Does not indicate when Fourier components are present in time In speech and audio processing, it is important to know when certain frequencies occurred

4. 15 - 4 Signal Energy Instantaneous power: |f(t)| 2 Complex signal: |f(t)| 2 = f(t) f * (t) Real-valued signal: |f(t)| 2 = f 2 (t) Example: f(t) is a voltage across a 1-  resistor Energy of signal, f(t) Energy dissipated when voltage f(t) is applied to 1  resistor Parseval’s Theorem Choose domain in which it is easier to compute energy

5. 15 - 5 Example #1 f(t) = e - a t u(t) is real-valued and causal

6. 15 - 6 Example #2 Determine the frequency W (rad/s) such that energy contributed by frequencies   [0, W] is 95% of the total energy E f = 1 / (2 a) W is known as the effective bandwidth

7. 15 - 7 Average Signal Power Time average of signals with infinite energy Signal does not go to zero as time goes to infinity Time average of amplitude squared (mean-squared value) Square root is root mean squared (RMS) value of signal Power signals must have infinite duration Periodic signals are power signals if finite energy in period Random signals are power signals Power of energy signal is zero Energy of power signal is infinite

8. 15 - 8 Examples Lathi, example 1.1 Finite Energy E f =8 Infinite energy but finite power P f = 1/3