Title: SIGNALS AND SPECTRA
1 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.
2Properties of Signals Noise
- 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
- ..
3Physically Realizable Waveforms
- 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 ajb,
where b is nonzero.
4Physically Realizable Waveforms
- 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.
Physical Waveform
Mathematical Model Waveform
- The Math model in this example violates the
following rules - Continuity
- Finite duration
5Time 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
6Periodic Waveforms
- Definition
- A waveform w(t) is periodic with period T0 if,
- w(t) w(t T0) for all t
- where 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, which - may be taken to be zero.
7DC Value
- Definition 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
8Power
- 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
9Evaluation 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)
Voltage
DC Value of this waveform is
Current
Instantenous Power
p(t) v(t)i(t)
10Evaluation of Power
The instantaneous power is
The Average power is
Maximum Power
Average Power
The Maximum power is PmaxVI
11RMS 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.
12Normalized Power
- In the concept of Normalized Power, R is assumed
to be 1?, 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
13Energy and Power Waveforms
- Definition w(t) is a power waveform if and only
if the normalized average power P is finite and
nonzero (0 lt P lt 8). - 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 lt E lt 8).
14Energy 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. (w(t) e-t). - Physically realizable waveforms are of the energy
type. - We can find a finite power for these!!
15Decibel
- 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
16Decibel Gain
- If resistive loads are involved,
Definition of dB may be reduced to,
or
17Decibel Signal-to-noise Ratio (SNR)
- Definition. The decibel signal-to-noise ratio
(S/R, SNR) is
Where, Signal Power (S)
And, Noise Power (N)
So, definition is equivalent to
18Decibel with Mili watt Reference (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
19Phasors
- 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 cej?c and Re. denotes
the real part of the complex quantity .. - The phasor can be written as