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

dtT

T

TT

2/

2/

1lim

ECE 4710: Lecture #3 6

Time Averages

If waveform is periodic (power) then becomes

where To is waveform period (fo = 1 / To ) DC value for power waveform:

For physical (energy) waveform:

dtT

T

T

2/

2/0

0

0

1

dttwT

WT

TDC

2/

2/0

0

0

)(1

dttwtt

Wt

tDC 2

1

)(1

12

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)

)()()( titvtp

)()()( titvtpP

dttwtt

dttwT

twWt

t

T

RMS 2

1

0)(

1or)(

1)( 2

120

2

0

2

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

RMSRMSRMSRMS IVRIR

VRti

R

tvP 2

22

2

)()(

dttwT

twIVPT

TRMSRMS

2/

2/

2

0

222 0

0

)(1

)(

dttwE

)(2

ECE 4710: Lecture #3 9

Example

Find DC,RMS value, Energy/Power for followingvoltage waveform:

Energy or Power? non-periodic Energy

Interval for integration? choose t = 0 to 4 (arbitrary)

V32824

1

|2|4|24

1)(

4

1)(

1 43

31

10

4

012

2

1

tttdttwdttwtt

Wt

tDC

0 1 2 3 4 t

4 V

2 V

ECE 4710: Lecture #3 10

Example

RMS value:

Energy:

V16.3|4|16|44

1

)(4

1)(

1)(

43

31

10

4

0

22

12

2 2

1

ttt

dttwdttwtt

twWt

tRMS

J40|4|16|4

)()(

43

31

10

4

0

22

ttt

dttwdttwE

ECE 4710: Lecture #3 11

Decibel

Decibel (dB) is base 10 logarithmic measure of power ratios

Relative measurement, e.g. Pout is 20 dB larger than Pin

Does not indicate actual magnitude of power level Must have reference power level to determine absolute power level

dB/1010&log10log10dB

i

o

i

o

in

out

P

P

P

P

P

P

ECE 4710: Lecture #3 12

S/N Ratio

Decibel Signal to Noise Ratio

)(

)(log10log10)/(

2

2

dBtn

ts

P

PNS

N

S

RMS

RMS

RMS

RMS

RMS

RMS

N

S

N

S

N

S

V

V

V

VNS

R

V

R

tn

R

V

R

ts

log20log10/

then)(

ispower noise

and )(

ispower signal Since

2

2

dB

22

22

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 (1•10 -3)

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 (1•10

-12) reference noise level » 0 dBrn = -90 dBm » Used in telephone industry

3101

(W) LevelPower log10dBm

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.

W/W2,50010or dB 34μW100

mW 250log10)/(

mW 25050

)54.3(

50

μW100)10(101dBm10 V54.32

5

34/10dB

22

10/103

NS

VS

NV

RMS

RMS

So

oS

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

jtjo

ootj

tjtjo

eAet( A w(t)

tjte

eet

o

o

oo

Re)cos if

)sin()cos(

)()cos(21

AAet( A w(t) jo or )cos