Signal Energy

1. Signal Energy

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. 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. 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 voltagef(t) is applied to 1W resistor • Parseval’s Theorem Choose domain in which it is easier to compute energy

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

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

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. Examples • Lathi, example 1.1 • Finite Energy • Ef=8 • Infinite energy • but finite power • Pf= 1/3