Signals and Spectra
Properties of Signals and Noise
In communication systems, the received waveform is usually categorized into the desired part containing the
information and the extraneous or undesired part. The desired part is called the signal, and the undesired
part is called noise.
Physically Realizable Waveforms
1.
The waveform has significant nonzero values over a composite time interval that is finite.
2.
The spectrum of the waveform has significant values over a composite frequency interval that is finite.
3.
The waveform is a continuous function of time.
4.
The waveform has finite peak values.
5.
The waveform is a Real function.
Reasons
1.
There is no known waveform that has existed forever.
2.
All practical systems have limited bandwidth which would no allow for infinite bandwidth.
3.
A discontinuity in the function would require infinite bandwidth.
4.
Any physical device would be destroyed before producing an infinite peak value.
5.
Only real waveforms have been observed in the real world.
Example
Important Properties and Definitions
Time Average Operator
One operator is linear if:
Periodic Waveform
for all t.
is the smallest positive number that satisfies this relationship.
Physical waveforms can not be truly periodic.
If the waveform is periodic the time average operator can be reduced to
Dc value
The dc value of a waveform
is given by its time average.
The expression has to be modified to the interval of interest.
Power
In communication systems, if the received (average) signal power is sufficiently large compared to the average noise
power, information may be recovered. Consequently, average power is an important concept that needs to be exactly
understood.
From circuit theory, the instantaneous power is:
p(t )  v(t )i (t )
Power
And the average power is:
p(t )  v(t )i(t )
RMS Value and Normalized Power
Definition.
Wrms 
w2 (t )
In electric circuits, for sine waves, when voltage and current are in phase,
Vrms  V / 2 ; I rms  I / 2
P
v 2 (t )
R

V 2 rms
R
P  I 2 rms R  Vrms I rms
Normalized power, R  1 .
P  w 2 (t )
Where w(t ) could be voltage or current.
Example
A 120 V, 60 HZ fluorescent lamp has unity power factor.
Find the DC voltage, the instantaneous power and the
average power.
Energy and Power Waveforms
w(t ) is a power waveform if the normalized average power P is finite and nonzero.
w(t ) is an energy waveform if the normalized energy E is finite and nonzero.
Decibel
This can also be expressed in terms of voltage or current.
If input and load resistances have the same value, or if this fact is disregarded,
The decibel signal-to-noise ratio
Other Decibel Meassures
Pin
1 mW
1W
1 kW
dB
dBm
dBW
dBk
Vector Space
A vector space V is a set that is closed under finite vector addition and scalar multiplication. The basic example is ndimensional Euclidean space
, where every element is represented by a list of n real numbers, scalars are real
numbers, addition is componentwise, and scalar multiplication is multiplication on each term separately.
Basis
A basis of a vector space V is defined as a subset
space span V. Consequently, if
every
of vectors in V that are linearly independent and vector
is a list of vectors in V, then these vectors form a basis if and only if
can be uniquely written as
where
, ...,
are elements of or . A vector space V will have many different bases, but there are always the
same number of basis vectors in each of them. The number of basis vectors in V is called the dimension of V. Every
spanning list in a vector space can be reduced to a basis of the vector space.
The span of subspace generated by vectors
and
is
Complete Basis
A set of orthogonal functions
is termed complete in the closed interval
continuous function f(x) in the interval, the minimum square error
if, for every piecewise
(where
denotes the L2-norm with respect to a weighting function w(x)) converges to zero as n becomes infinite.
Symbolically, a set of functions is complete if
where the above integral is a Lebesgue integral.
Examples of complete orthogonal systems include
over
(which actually form a slightly
more special type of system known as a complete biorthogonal system
Orthogonal Series Representation of Signals and Noise
Orthogonal Functions
Coefficients
Complex Fourier Series
Complex Fourier Series
Quadrature Fourier Series
Polar Fourier Series