ECE 4710: Lecture #3 1 Signals & Noise  Received waveform contains the desired signal and the undesired noise  We will use deterministic waveforms (not random) to mathematically model signals  Important signal and noise properties  DC and RMS levels  Average Power  Bandwidth  Magnitude and Phase Spectrums  Power Spectral Density (PSD)

ECE 4710: Lecture #3 2 Signal Properties  Received waveform is an electrical signal and is either a time-varying voltage, v(t), or current, i(t)  Book uses general notation of w(t) to represent either voltage or current  Properties of physically realizable waveforms:  Significant amplitudes over finite time interval »Real signals and systems exist for finite amount of time  Significant spectral amplitudes over finite frequency interval »Any channel (coax, wireless, etc.) has finite BW

ECE 4710: Lecture #3 3 Signal Properties  Properties of physically realizable waveforms:  Waveform is continuous function of time  Waveform has finite peak values »Physical devices destroyed with infinite peak values  Waveform has only real values »Complex math used to represent signal properties such as phase  Math models that violate some of these conditions can and will be used  Simplifies analysis and yields  correct results if done properly

ECE 4710: Lecture #3 4  Math waveform has infinite bandwidth due to discontinuous time  Average power of real and math waveforms same Signal Properties

ECE 4710: Lecture #3 5 Time Averages  Time average operator,  , is  Power and Energy Signals  Power: periodic signals from -  to +  with infinite energy »Not physically realizable but useful model over finite time intervals  Energy: non-periodic (aperiodic) signals with finite energy »Use power signals over finite time to model

ECE 4710: Lecture #3 6 Time Averages  If waveform is periodic (power) then   becomes where T o is waveform period ( f o = 1 / T o )  DC value for power waveform:  For physical (energy) waveform:

ECE 4710: Lecture #3 7 Power  If average received signal power is sufficiently larger than average received noise power  information may be recovered (imperfectly)  Instantaneous power:  Average power:  Root Mean Square (RMS) value of w (t)

ECE 4710: Lecture #3 8 Power & Energy  For resistive electrical circuits  Normalized Power  R = 1  then  Normalized Energy  Energy waveform if 0 < E < , otherwise power waveform

ECE 4710: Lecture #3 9 Example Find DC,RMS value, Energy/Power for following voltage waveform: Energy or Power?  non-periodic  Energy Interval for integration?  choose t = 0 to 4 (arbitrary) t 4 V 2 V

ECE 4710: Lecture #3 10 Example RMS value: Energy:

ECE 4710: Lecture #3 11 Decibel  Decibel (dB) is base 10 logarithmic measure of power ratios  Relative measurement, e.g. P out is 20 dB larger than P in  Does not indicate actual magnitude of power level  Must have reference power level to determine absolute power level

ECE 4710: Lecture #3 12 S/N Ratio  Decibel Signal to Noise Ratio

ECE 4710: Lecture #3 13 dBm  Decibel can be used to indicate absolute power level if reference power is used  “m” used to denote the mW reference level ( )  0 dBm = 1 mW 30 dBm = 1 W  Other reference levels also used:  dBW uses 1 W reference level  0 dBW = 1 W  dBrn uses 1 pW ( ) reference noise level »0 dBrn = -90 dBm »Used in telephone industry

ECE 4710: Lecture #3 14 Example  A signal voltage of 5 cos(  t) is measured at the output of a communication receiver across a 50  load resistor. If the output noise is measured to be –10 dBm, find the output S/N ratio in dB and W / W.

ECE 4710: Lecture #3 15 Phasors  Phasor notation used to represent sinusoidal waveforms  cosine chosen for reference  Shorthand notation  frequency implicit and Re is dropped so