ebb2073 1 chapter 2 signals and spectra chapter objectives: basic signal properties (dc, rms, dbm,...

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Chapter 2

SIGNALS AND SPECTRA

Chapter Objectives: • Basic signal properties (DC, RMS, dBm, and power); • Fourier transform and spectra;• Linear systems and linear distortion;• Band limited signals and sampling;• Discrete Fourier Transform;• Bandwidth of signals.

Dr. Azlan Awang, EBB2073

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Properties of Signals & Properties of Signals & NoiseNoise

In communication systems, the received waveform is usually categorized into two parts:

Signal:The desired part containing the

information.

Noise:The undesired part

Properties of waveforms include:

• DC value, • Root-mean-square (rms) value, • Normalized power, • Magnitude spectrum,

• Phase spectrum, • Power spectral density, • Bandwidth • ………………..


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Physically Realizable Physically Realizable WaveformsWaveforms

Physically realizable waveforms are practical waveforms which can be measured in a laboratory.

These waveforms satisfy the following conditions

• The waveform has significant nonzero values over a composite time interval that is finite.

• The spectrum of the waveform has significant values over a composite frequency interval that is finite

• The waveform is a continuous function of time• The waveform has a finite peak value• The waveform has only real values. That is, at any time, it

cannot have a complex value a+jb, where b is nonzero.


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Physically Realizable Physically Realizable WaveformsWaveforms

Mathematical Models that violate some or all of the conditions listed above are often used.

One main reason is to simplify the mathematical analysis. If we are careful with the mathematical model, the correct

result can be obtained when the answer is properly interpreted.

The Math model in this example violates the following rules:

1. Continuity2. Finite duration

Physical Waveform

Mathematical Model Waveform


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Time Average Operator

Definition: The time average operator is given by,

The operator is a linear operator, • the average of the sum of two quantities is the same as

the sum of their averages:


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Periodic WaveformsPeriodic Waveforms Definition

A waveform w(t) is periodic with period T0 if,

w(t) = w(t + T0) for all twhere T0 is the smallest positive number that satisfies this relationship

A sinusoidal waveform of frequency f0 = 1/T0 Hertz is periodic

Theorem: If the waveform involved is periodic, the time average operator can be reduced to

where T0 is the period of the waveform and a is an arbitrary real constant, whichmay be taken to be zero.


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DC ValueDefinition: The DC (direct “current”) value of a

waveform w(t) is given by its time average, w(t). Thus,

For a physical waveform, we are actually interested in evaluating the DC value only over a finite interval of interest, say, from t1 to t2, so that the dc value is


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PowerPower

Definition. Let v(t) denote the voltage across a set of circuit terminals,

and let i(t) denote the current into the terminal, as shown .

The instantaneous power (incremental work divided by incremental time) associated with the circuit is given by:

p(t) = v(t)i(t)

the instantaneous power flows into the circuit when p(t) is positive and flows out of the circuit when p(t) is negative.

The average power is:Dr. Azlan Awang, EBB2073

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Evaluation of DC ValueEvaluation of DC Value A 120V , 60 Hz fluorescent lamp wired in a high power factor configuration.

Assume the voltage and current are both sinusoids and in phase ( unity power factor)

0

0

0

/ 2

0/ 20

0 0

0 0

( ) cos

1cos 0

Where,

2 / ,and

1/ 60

Similarly, 0

DC

T

T

DC

V v t V t

V t dtT

T

f T Hz

I

DC Value of this waveform is:

Instantenous Power

Current

Voltage

p(t) = v(t)i(t)Dr. Azlan Awang, EBB2073

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Evaluation of PowerEvaluation of Power

0

0

0

20

20

1/ 2 1 2cos

1 2

2

cos2

T

T

P VI t

VIt dt

T

VI

The instantaneous power is:

The Average power is:

tVI

tItVtp

0

00

cos212/1

coscos)(

The Maximum power is: Pmax=VI

Average Power

Maximum Power


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RMS ValueRMS Value Definition: The root-mean-square (rms) value of w(t) is:

Rms value of a sinusoidal:

Theorem: If a load is resistive (i.e., with unity power factor), the average

power is:

where R is the value of the resistive load.

2cos( )

2rms o

VW V t


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Normalized PowerNormalized Power In the concept of Normalized Power, R is assumed to be 1Ω,

although it may be another value in the actual circuit. Another way of expressing this concept is to say that the

power is given on a per-ohm basis. It can also be realized that the square root of the normalized

power is the rms value.

Definition. The average normalized power is given as follows, Where w(t) is the voltage or current waveform


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Energy and Power WaveformsEnergy and Power Waveforms

Definition: w(t) is a power waveform if and only if the normalized average power P is finite and nonzero (0 < P < ∞).

Definition: The total normalized energy is

Definition: w(t) is an energy waveform if and only if the total normalized energy is finite and nonzero (0 < E < ∞).


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Energy and Power WaveformsEnergy and Power Waveforms If a waveform is classified as either one of these types, it

cannot be of the other type.

If w(t) has finite energy, the power averaged over infinite time is zero.

If the power (averaged over infinite time) is finite, the energy if infinite.

However, mathematical functions can be found that have both infinite energy and infinite power and, consequently, cannot be classified into either of these two categories. (E.g., w(t) = e-t).

Physically realizable waveforms are of the energy type. – We can find a finite power for these!!


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DecibelDecibel

A base 10 logarithmic measure of power ratios. The ratio of the power level at the output of a circuit

compared with that at the input is often specified by the decibel gain instead of the actual ratio.

Decibel measure can be defined in 3 ways• Decibel Gain

• Decibel signal-to-noise ratio

• Mill watt Decibel or dBm

Definition: Decibel Gain of a circuit is:


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Decibel GainDecibel Gain

If resistive loads are involved,

Definition of dB may be reduced to,

or


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Decibel Signal-to-noise Ratio Decibel Signal-to-noise Ratio (SNR)(SNR)

Definition. The decibel signal-to-noise ratio (S/R, SNR) is:

Where, Signal Power (S) =

So, definition is equivalent to

And, Noise Power (N) =


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Decibel with Mili watt Reference Decibel with Mili watt Reference (dBm)(dBm)

Definition. The decibel power level with respect to 1 mW is:

= 30 + 10 log (Actual Power Level (watts)

• Here the “m” in the dBm denotes a milliwatt reference. • When a 1-W reference level is used, the decibel level is

denoted dBW;• when a 1-kW reference level is used, the decibel level is

denoted dBk.

E.g.: If an antenna receives a signal power of 0.3W, what is the received power level in dBm?dBm = 30 + 10xlog(0.3) = 30 + 10x(-0.523)3 = 24.77 dBm


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Definition: A complex number c is said to be a “phasor” if it is used to represent a sinusoidal waveform. That is,

where the phasor c = |c|ejc and Re. denotes the real part of the complex quantity ..

The phasor can be written as:

PhasorsPhasors

jc x jy c e c


ebb2073 1 chapter 2 signals and spectra chapter objectives: basic signal properties (dc, rms, dbm,...

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