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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.
The waveform has significant nonzero values over a composite time interval that is finite. There is no known waveform that has
existed forever. The spectrum of the waveform has
significant values over a composite frequency interval that is finite. All practical systems have limited bandwidth
which would no allow for infinite bandwidth.
The waveform is a continuous function of time. A discontinuity in the function would
require infinite bandwidth. The waveform has finite peak values.
Any physical device would be destroyed before producing an infinite peak value.
The waveform is a Real function. Only real waveforms have been
observed in the real world.
Important Properties and Definitions Time Average Operator
This operator is linear:
Important Properties and Definitions
Periodic Waveform for all values of t. is the smallest positive number that satisfies this relationship.
If the waveform is periodic the time average operator can be reduced to
Physical waveforms cannot be truly periodic.
Important Properties and Definitions
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:
)()()( titvtp
Power And the average power is:
)()()( titvtp
RMS Value Definition.
)(2 twWrms
Normalized Power In electric circuits, for sine waves, when voltage and current are in phase,
2/VVrms
; 2/IIrms
R
V
R
tvP
rms22 )(
rmsrms
rms IVRIP 2
Normalized power, 1R . )(2 twP
Where )(tw 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
)(tw is a power waveform if the normalized average power P is finite and nonzero.
)(tw 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,
Decibel
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 dB 1 mW dBm 1 W dBW
1 kW dBk
Basis A basis of a vector space V is defined as a subset of vectors in V that
are linearly independent and vector space span V. Consequently, if is a list of vectors in V, then these vectors form a basis if and only if every 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
Vector Space A vector space V is a set that is closed under finite vector addition and scalar multiplication. The basic example is n-dimensional Euclidean space , where every element is represented by a list of n real numbers, scalars are real numbers, addition is component wise, and scalar multiplication is multiplication on each term separately.
Vector Space
A set of orthogonal functions is termed complete in the closed interval
if, for every piecewise continuous function f(x) in the interval, the minimum square error
(where denotes the L2-norm with respect to a weighting function w(x)) converges to zero as n becomes infinite.
Orthogonal Series Representation of Signals and Noise
Orthogonal Functions
Coefficients
Complex Fourier Series
Rectangular and Triangular Pulses
Rectangular and Triangular Pulses
Rectangular and Triangular Pulses
Complex Fourier Series and Line Spectra
0
0
00
00
00
0
0
)(
)(
)(
)(
)(
)(
)(
2
22
)22(
)22(
)(2
2
nffn
nff
ftj
n
n
tnfjtnfj
nd
n
tnfdtnfj
nd
n
tnftnfj
nd
n
ttnfj
nd
n
tnfj
n
fTAdSac
efTAdSac
eedttw
edttw
edttw
edttw
edtw
d
d
dn
d
d
Complex Fourier Series
Complex Fourier Series
n
Quadrature Fourier Series
Polar Fourier Series
Polar Fourier Series
Power Spectral Density
