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Sine waves and Decibels
Southern Methodist University
EETS8302
Fall 2005
Lecture 3
Typographic error on p.11 corrected
Slides only. (No notes.)
Rev.5;Page. 1
©1996-2005 R.Levine
PSTN Transmission Technology in 1960
• Analog transmission of voice channel signals
– Analog amplifiers (“repeaters”) inserted in long transmission
lines compensate for primarily resistive losses
– Analog frequency division multiplexer (FDM) used for long
distance
– Shortcomings of analog amplifiers, even with the use of
“negative feedback”:
» Some distortion “peak flattening” of waveforms
» Some random noise added by each amplifier
• Transistors (vis-à-vis vacuum tubes) slightly extended
the useful life of analog technology
• From 1960 to today (2005) digital multiplexing
transmission technology has almost completely
replaced analog transmission
• Widespread rapid growth of digital multiplexing (e.g. T1 or E-1) facilitated the introduction of digital switching
Rev.5;Page. 2
©1996-2005 R.Levine
PSTN Switching Technology in 1960
• Dial-up calls switched via electromechanical
“space-division” switches
– Old Strowger step-by-step “(“gross motion”) switches
– Newer “crossbar” (“fine motion”) electromechanical
switches. Intended evolution of switching until eventually
superseded by digital switching
– Then new computer controlled analog switches (example
AT&T 1ESS – electronic switching system)
» Quite sophisticated control and call processing
» Switching used small “reed relay” metallic contacts
• From 1960 to today (2005) digital switching
has almost totally replaced analog
Rev.5;Page. 3
©1996-2005 R.Levine
Data Communication in 1960
• Modulator-demodulators (MODEMs) using frequencyshift-keying (FSK) modulation were in limited use via
dial-up PSTN connections
– Data rates up to 110 bit/second typically used with
electromechanical “teletypewriter exchange” (TWX) services via
voice channels.
– TWX was an AT&T competitor to the Western Union TELEX
service. Telex used special electromechanical switches and a
separate private network. TWX and Telex have since been
substantially supplanted by Internet e-mail.
• Higher modem bit rates (<2400 bits/second) typically
required a leased point-to-point line with a fixed
“equalizer” (analog waveform compensation to correct
for waveform changes due to transmission
impairments)
– Costly and slow-response installation requiring skilled
technicians
– Self-adaptive equalizers opened the dial-up market to higher bit
rate data transmission for computers, etc.
Rev.5;Page. 4
©1996-2005 R.Levine
Legacy Methods and Jargon
• The subscriber loop and telephone set still use analog
waveforms in the vast majority of installations today
• Speech or test waveform power level is described
using decibels (dB)
– Proportional to common logarithm of the ratio of the power
compared to a 1 milliwatt (mW) reference signal
– Historically introduced about 1910. One dB described the inputoutput power ratio of a 1 kHz test tone in a mile (1.6km) of 19 ga
copper wire*.
– Calculation of total “loss” (output power/ input power) for
cascaded transmission equipment (e.g., several miles of wire)
requires adding dB values, rather than multiplying power ratios.
• Unlike a MODEM, the human ear is not sensitive to
waveform changes caused by varying delay of different
frequency components of a waveform. Therefore,
measurement of signal power vs. test sine wave
frequency is sufficient to characterize an analog
telephone channel.
*North American wire gauge units (ga or AWG) explained later.
Rev.5;Page. 5
©1996-2005 R.Levine
Sine Wave Test Signals
• Analog telephone channels, and end-to-end
performance of analog channels with a digitally coded
transmission segment in the middle, can be
characterized via:
– Signal power output-input ratio (expressed in dB) vs. frequency
(Hz), using a sine wave test signal
» For example, power loss must be uniform within limits
stated in ITU standards over the frequency range 300 Hz to
3500 Hz.
– For MODEM use, the test signal time delay vs. frequency
» Alternatively to time delay: phase shift vs. frequency.
• Above information typically described via a two-axis
graph or a table (list)
• Any non-sinusoidal waveform can theoretically be
made up of a sum of various properly chosen sine
waves of different amplitudes and frequencies.
– So-called “Fourier analysis and synthesis”
– Analysis of arbitrary waveform results can be theoretically
computed with relative ease
Rev.5;Page. 6
©1996-2005 R.Levine
Example Voice Channel Power
v. Frequency Graph
dB
This is derived from the
amplitude vs. frequency
Measurement.
0
-3
-3 dB (“half power”) point is used only
because it is convenient for measurements.
-10
300 Hz low frequency signal drop off is result
of coupling transformers in the subscriber line
circuit. 3500 Hz high frequency signal drop off
is result of intentional low pass filter design, to
retain adequate signal quality for a conversation.
-20
-30
kHz
0
300
Rev.5;Page. 7
1
2
©1996-2005 R.Levine
3
3500
4
Describing Linear Systems
• Many telecommunications transmission system
elements are accurately described by linear equations.
• Example linear equation: y= K•x, where y and x are
measurable system variables like voltage and current
and K is a constant.
• Another example: y= K•(dx/dt), showing that a time
derivative is a linear “operation”
• Many electrical devices like resistors, capacitors,
inductors and transmission lines (wire, cable) are
inherently linear.
• Some devices, like amplifiers, are non-linear overall,
but are intentionally designed and used in only a part
of their range of voltage and current where they appear
to be linear.
Rev.5;Page. 8
©1996-2005 R.Levine
Graphical Representations
y
v
(0,0)
x
vout
i
vo
Linear range
vin
Rev.5;Page. 9
©1996-2005 R.Levine
vi
Linear System Analysis
• Linear devices and systems can
be described for purposes of
analysis or design, by either of
two methods:
– Time domain analysis: output waveform,
described as graph/table/list of voltage vs.
time, in response to ideal impulse input
– Frequency domain analysis: two
graphs/tables/lists showing amplitude vs.
frequency and phase (time delay) vs.
frequency) in response to input sine wave
at many test frequencies
Rev.5;Page. 10
©1996-2005 R.Levine
Symbolic: pulse
time very small,
Pulse voltage
very big.
Ideal Unit Impulse
1 v•s
v
t
t
• The unit impulse is a theoretical waveform that is zero
amplitude for all time before and after time t=0.
– At time t=0, it is “infinitely” large, so that its “area’ when drawn
on a graph is unity (1).
– For the case where the impulse is a voltage, its “area” is stated
as one volt-second. When the impulse is a fluid flow quantity like
liters/second, its “area” is stated as one liter.
– Symbols used for this waveform are: u0(t) or (t) or I(t).
• The time integral of a unit impulse is a unit step.
– For real-life measurement of the impulse response, measure or
“capture” the step response, then compute the time derivative of
the step response. This derivative waveform, treated as a
voltage waveform is the impulse response.
Note: A volt-second is also called a weber (Wb).
Rev.5;Page. 11
©1996-2005 R.Levine
Time Domain Analysis Example
input
output
v
Impulse response
System
v
t
v
input
t
t
v
t
Rev.5;Page. 12
To compute output for an
arbitrary input, cut input
waveform into small pieces
and replace each piece with
an impulse of equal area.
Each input impulse produces
a proportional and delayed
impulse response. Add all
of these. A formal method
to do this is called the
convolution integral (calculus).
Approximate output waveform made up of
sum of several copies of the impulse response
(delayed and scaled to the input signal). For
theoretical analysis we find the result of infinitely
many very closely spaced impulse responses.
©1996-2005 R.Levine
Frequency Domain Analysis
• Using a sine wave generator of unit amplitude, feed a
sine waveform into the system at every frequency.
Theoretically this requires an infinite number of
distinct frequency choices.
• For each input sine wave (actually a cosine wave)
frequency, the output will also be a sine wave having
the same frequency but in general a different amplitude
and phase (time delay) relative to the input sine wave.
• This information is compiled into two lists, amplitude
vs. frequency and phase vs. frequency.
• When an input waveform is already properly described
by means of a similar pair of lists, compute the
frequency domain description of the output by means
of these operations:
– The amplitude of the output signal is the product (multiply) of
the input amplitude and the system amplitude.
– The phase of the output signal is the sum (add) of the phase of
the input and the phase of the system.
Rev.5;Page. 13
©1996-2005 R.Levine
Comparison
• The multiplication (of amplitudes) and
addition (of phases) used in frequency
domain analysis are simpler than the use of
integral calculus to compute the convolution
integral in time domain analysis.
• But we ultimately live in the time domain, so
to determine the output waveform somebody
somehow somewhere must do some integral
calculus.
– Books are available containing tables of “transforms”
listing the frequency domain description (via table,list,
graph or formula) for many well-know time domain
waveforms.
Rev.5;Page. 14
©1996-2005 R.Levine
Time-Frequency Domain Relationships
• The two methods are theoretically related
• The unit impulse waveform can be shown to be equal
to the sum of infinitely many cosine waves (each of
unit amplitude).
• In general, each time domain input waveform can be
“transformed” (described in the form of a frequency
domain list/table/formula). The frequency domain data
for many well-known waveforms (step, triangular
waveforms, etc.) are given in mathematical handbooks
and treatises, or they can be computed by means of
well-known methods or via numerical calculation using
digital computers.
• This process of converting from a waveform in time
domain to a description in the frequency domain is
called a Fourier transform, after its inventor.
• A similar process, called LaPlace transformation, is not
described here.
Note: Fourier (a French name) is pronounced /four-yay/
Rev.5;Page. 15
©1996-2005 R.Levine
Why Engineers Love Sine Waves*
• Linear systems can be analyzed relatively simply for sine wave
signals using complex number algebra
– Differential/integral equations are not needed explicitly
– Sine waves and “real” exponential waveforms are “natural behavior”
waveforms of linear systems
• The natural oscillations of linear time-invariant electrical (and
mechanical) systems comprise
– Sine/cosine waves
– “Real” exponential waveforms
– Product combinations of these two
• Many systems are designed to use sine waves partly because of
ease of analysis:
– Electric power systems use 50 Hz or 60 Hz sine waves (400Hz on aircraft).
» Originally, electrically linear devices like incandescent lights, motors, etc. were
the only devices in power networks. Today we get harmonic power waveform
distortion from fluorescent lights and other non-linear electronic equipment!
» Radio broadcasting carrier waveforms
* Many engineers also have passionate feelings for electrons, as well!
Rev.5;Page. 16
©1996-2005 R.Levine
More Sine Wave Features
• Fourier Series: Any periodic (cyclic repetitive)
waveform can be mathematically expressed as a
discrete sum of (possibly an infinite series of) sine
waves.
– The base frequency of the series of sine waves is the same as
the repetition frequency of the waveform to be described
– The waveform can have step discontinuities in voltage, sharp
impulses, other bizarre mathematical features…
• Fourier Integral: Any non-repetitive waveform can be
mathematically expressed as an integral (summation of
an uncountable continuum) of sine waves at all
frequencies.
– Again discontinuous and impulsive waveforms are allowable
• Distinct radio transmissions are separated from each
other by means of a transmitter and receiver that
purposely produce a radio waveform of limited
sinusoidal waveform bandwidth and a receiver that
amplifies and processes (detects) only a
corresponding limited portion of the radio frequency
sine wave spectrum.
Rev.5;Page. 17
©1996-2005 R.Levine
Natural Waveforms of Linear
Systems
• Sine wave
• Exponential
• Product of the two
5
4
3
2
1
v( t ) 0
1
2
3
4
5
1
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
2.7. sin 2.  . ( 5 ) . t
v( t )

t
1
0.8
8
0.6
a( t )
0.4
a( t )
exp
t
0.2
0
0.3
4
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
t
2
b( t )
exp
t . .

2.7sin 2. .( 5).t
0.3
8
b( t )
0
2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
t
Both waveforms are subsets of the class of complex exponential mathematical functions.
Rev.5;Page. 18
©1996-2005 R.Levine
What is a Linear System?
• Consider physical systems having “input” and
“output” waveforms
– Waveforms may be current, voltage, mechanical variables like
velocity, current, force, etc. [v(t), i(t), f(t), etc.]
– Each input waveform produces a distinct output waveform
• Consider a compound input waveform made up of the
sum of two other waveforms
• In a linear system, the “output” due to such a
compound “input” is the sum of the outputs due to
each input acting alone
– Sometimes called the “principle of superposition”
• Note: Do not confuse this technological jargon
meaning of “linear” with the currently popular
alternative meaning of “sequential” or “consecutive”
Rev.5;Page. 19
©1996-2005 R.Levine
Properties of Linear Systems
• Multiplying the input amplitude by a constant number
factor m causes the output to increase in amplitude by
the same factor m
– If input waveform f(t) produces output waveform g(t), then higher
amplitude waveform m•f(t) produces higher amplitude output
waveform m•g(t)
• A system described by linear algebraic equations with
constant coefficients is a time-invariant linear system:
z = A•x + B•y
w= C•x + D•y
If A,B,C, D are not constants, but vary with time, but do not depend on w,x,y or z,
the system is still linear, but not with constant coefficients . Linear time-varying
systems still follow the principle of superposition, but their “natural” behavior
waveforms may not be sine waves
• A system accurately described by linear differential or
integral equations is also linear:
example, a capacitor: i = C•dv/dt
Rev.5;Page. 20
©1996-2005 R.Levine
•
Many Actual Systems are
Intended
to
be
Linear
Electric circuits consisting of resistors, capacitors,
inductors, transformers and transmission lines are
linear when used within their design range
– However, inductors and transformers can be “over driven” into a
non-linear regions of current due to non-linear magnetic
properties of some materials, like iron (careful!)
– If resistors get hot enough to burn or melt, they no longer have
“constant coefficients” of resistance
• Most electronic devices like diodes, transistors, etc.,
are very non-linear
– When “small signals” (low amplitude) are used, their properties
are approximately linear
• Many electronic systems which are not precisely linear
can be approximately described as linear over a
restricted range of voltage or current.
Rev.5;Page. 21
©1996-2005 R.Levine
A Sine Wave
• The “shadow” or projection onto one axis, of a point
moving in a circle in 2-dimensional space
sin 2.  . f1. t
v( t )
•
1
1
v( t ) 0
1
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
t
Here, f1 is 1 Hz, amplitude is 1, period is 1 second, initial phase angle is zero.
Rev.5;Page. 22
©1996-2005 R.Levine
Three Sine Wave Properties
• Frequency
– common unit: cycles per second or hertz (Hz)
– Other units: degrees per second, radians per second, etc. (360
degrees or 2• = 6.28... radians in one cycle or turn)
– Period or cycle duration is the reciprocal of frequency: T=1/f
• Amplitude
– Unit: the physical quantity which the sine wave represents;
amperes (current), volts (voltage), etc.
• Phase (angle) or time delay
– relative delay or time offset compared to a reference waveform or
clock signal
– Unit: expressed in degrees, radians, fractions of a cycle, or in
time units (seconds, etc.)
Rev.5;Page. 23
©1996-2005 R.Levine
Non-smooth appearance
of sine wave is an artifact
of the drawing software.
5
4
3
2
1
v( t ) 0
1
2
3
4
5
Example
amplitude
v( t )
2.7. sin 2.  . ( 5 ) . t

8
frequency
phase
1
1
•
Peak
amplitude
2.7 volts
(corresponds
to 1.909 v RMS
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
phase delay compared
to cosine wave reference
is /8 radians, or 22.5º
or 1/16 cycle, or 0.0125 sec
or 12.5 milliseconds
t
period or interval is 0.2 sec,
or 200 milliseconds. Then
frequency is 1/(0.2) or 5 Hz.
Note: Sine wave with 90º phase advance is equivalent to a Cosine wave.
Rev.5;Page. 24
©1996-2005 R.Levine
Sine Waves in Linear Systems
• Sine wave input to a linear time invariant system produces
sine wave output (in “sinusoidal steady state”)
– In many cases there is also a “transient” or start-up response
(decreasing exponential) waveform which eventually “dies out”
(amplitude becomes very small).
• Output is precisely the same frequency as the input
• The only output parameters that can differ from the input
waveform are:
– Physical value (current vs. voltage, for example)
– Amplitude
– Phase
• The amplitude and phase may be expressed symbolically in
complex number algebra (real and imaginary numbers) by
one two-part symbol on paper.
– “Real” ratios such as resistance (ohm=volt/amp) can be generalized to
a complex number called “impedance” which indicates both the
magnitude ratio and the phase angle difference as well.
Rev.5;Page. 25
©1996-2005 R.Levine
Why is Linearity Important?
• Non-linear systems, in general, produce signals which have
different sinusoidal frequency waveforms at the output than
the sine wave frequency used at the input
• Some systems exploit this property -- radio receivers
incorporate a “down converter” which produces internal radio
signals at a lower frequency than the radio input waveform at
the antenna. Lower frequency waveforms are often less
complicated to process and equipment is thus less costly.
• One such radio process: “detection” or “demodulation” -extracts original information waveform such as speech or
digital information.
• When other undesired frequencies are produced, this is called
Inter-Modulation (IM)
– When it occurs just as desired, it is called Modulation. This usually
involves intentionally generating a waveform which is the instantaneous
arithmetic product of two waveforms.
– The word “modulation” has several meanings, some rather vague.
Rev.5;Page. 26
©1996-2005 R.Levine
Ultra Wide Band
• The only propose radio signal competitor to using a
modulated sine wave is UWB, which proposes to use
approximately periodic extremely brief high amplitude
electromagnetic pulses
• Information is conveyed by varying the time interval
between the pulses
– This is sometimes called “pulse position modulation” PPM
• Proponents claim that the large rf bandwidth and the
low amount of UWB power in a typical narrower signal
bandwidth permits simultaneous operation without
mutual interference
– Some opponents dispute such claims in fundamentals
– Other opponents assert that the inter-modulation of UWB and
other ordinary bandwidth signals is inherently severe, and
causes serious interference to ordinary bandwidth signals.
• At this time, efforts are still underway to utilize UWB,
and not all parties agree.
Rev.5;Page. 27
©1996-2005 R.Levine
“Modulation” of a Square Wave
Fig. 1 PWM
T
Fig. 2 PPM
time
• A square wave (in contrast to a sine wave) is seldom modulated in
either frequency or amplitude in modern electronics, although that
is technologically feasible.
• Some equipment “modulates” the width of square pulses (PWM).
See Fig. 1
• Some equipment modulates the position of a pulse in the time
window T reserved for each period of pulse transmission (PPM).
Fig. 2. One type of PPM is “ultra-wide-band” radio.
– Incidentally, PPM waveform is the approximate time derivative of a corresponding
PWM waveform.
• We speak of “pulse code modulation” (PCM) where an analog
voltage is measured and the measured voltage is encoded as a
sequence of binary pulses (amplitude either 5 volts or zero volts).
PCM is unlike PWM or PPM.
©1996-2005 R.Levine
Rev.5;Page. 28
Sine Waves, Signals, and
Spectrum Analysis
Any waveform may be theoretically described as a
combination of sine and cosine waves of various
frequencies:
• Fourier series for periodic waveforms
– Frequencies are integral multiples of basic repetition frequency
• Fourier integral for non-periodic waveforms
– Theoretically infinite number of frequency components
• Other mathematical descriptions are used for special
purposes, as well:
*Also called
Taylor or
MacLaurin
Series.
– Power* series: y = ao + a1•t + a2•t2 + a3•t3 … is used in linear
predictive speech coders, for example.
– Power series coefficients ao,a1,etc. derived from nth time
derivative of waveform to be described...
Rev.5;Page. 29
©1996-2005 R.Levine
Linear systems are described two ways:
• Response to sine waves of (theoretically) all
frequencies
– List(s), graph(s)or other descriptions are used
– Describe amplitude and phase of output for each frequency
– Test input is a unit amplitude cosine wave (sine wave with 90º
phase advance) at each test frequency
• Impulse response (time domain)
– Output is a waveform produced by a theoretical input pulse
signal
– Input, in theory, is an instantaneous impulse of infinite amplitude
and unit “area” (in volt•sec, for example)
– For practical lab measurements on systems having a limited
linear range of operation, a unit step input voltage is used. The
time deerivative of the output is the impulse response.
• These two descriptions are theoretically mutually
derivable, either one from the other
Rev.5;Page. 30
©1996-2005 R.Levine
Sample Circuit with Resistors
and Capacitors
Adjustable
Frequency
Sine-wave
Test Signal
Generator
+
V1
--
+
V2
--
Voltage-ratio
meter, with
display showing
ratio as a function
of oscillatory
frequency.
Rev.5;Page. 31
©1996-2005 R.Levine
What to Display and Why...
• Computing, tabulating or plotting a graph of the ratio of
output/input voltage allows us to compute the absolute output
voltage for any specific input voltage waveform.
• Logarithmic graph axis displays a wide range of frequency (or
voltage or power).
– In certain types of electric devices, the slope of some asymptotic lines
(log voltage vs. log frequency) is exactly 1, 2, etc., indicating that voltage
is proportional to the frequency, the square of the frequency, etc.
• The unit scale is normally chosen so the labeled axes are
integral values such as 1, 2, 5, 10, 20, 50, 100 etc. The peculiar
numbers in the following examples are an artifact of the
graphics software, which needs to be improved!!
• Two times the log of the voltage ratio shows the power ratio (in
systems having the same ratio of voltage to current at input
and output). Twenty times the log of the voltage ratio in such a
case is called the decibel (dB) power ratio.
Rev.5;Page. 32
©1996-2005 R.Levine
Logarithms
• 18th century Scottish mathematician Napier noted that
when multiplying two numbers that were expressed as
powers of the same base, the exponents were simply
added. When dividing, the exponent of the divisor is
subtracted. Examples (using base 2):
• 32•16= 25 • 24= 2(5+4) = 29 = 512
• 512/128 = 29/27 = 2(9-7) = 22 = 4
• Napier computed and published a table of base-10 (socalled “common”) logarithms to reduce the labor
required for multiplying and dividing.
• Logarithms were extensively taught in schools before
the availability of electronic calculators. They have
many other mathematical uses as well.
– Logarithms are the basis of the “slide rule” which has also been
substantially supplanted by the electronic calculator or
computer.
Rev.5;Page. 33
©1996-2005 R.Levine
Phase vs. Frequency Graph
P hase Ang le vs. lo g (F req uency)
90
60
p hase (d eg )
30
0
1.00
P hase
34.66
1,201.12
41,627.70
-30
-60
-90
freq u en cy (H z)
Rev.5;Page. 34
©1996-2005 R.Levine
1,442,700.00
50,000,000.00
Amplitude Ratio (V2/V1) Graph
A m p litud e vs . lo g (F re q ue nc y)
1
0.9
0.8
0.7
Peak amplitude
~ 0.62
V ou t/V in
0.6
0.5
A m plitude
“Half-power” amplitude
0.44:= 0.62•0.707
0.4
0.3
Bandwidth
Based on
Half-power
points
0.2
0.1
0
1
34.6572
1201.12
41627.7
freq u en cy (H z)
Rev.5;Page. 35
©1996-2005 R.Levine
1442700
50000000
Gain [20 log(V2/V1)] Graph
20LogAmplitude (dB) vs. log(Frequency)
frequency
1.00
34.66
1,201.12
50,000,000.00
41,627.70
1,442,700.00
0.00
-20.00
-40.00
-60.00
-80.00
-100.00
-120.00
Rev.5;Page. 36
©1996-2005 R.Levine
G ain(dB)
Significance of These Graphs
• The previous graphs show that a simple electric circuit using
capacitors and resistors can perform the operation of frequency
“filtering” (not the best filter example, but simple to analyze)
• In this example, the output voltage (or power) is maximum for a
frequency of about 35 Hz.
– The output voltage is much lower (“more attenuated”) for higher or lower
frequencies.
– The output voltage is very small (in fact, zero although the logarithmic axes do not
show this) for zero frequency and “infinite” frequency.
– Such a “band pass” filter is used to separate one radio frequency from others so a
radio receiver can receive a desired broadcast or cellular conversation.
– Most band pass filters used in radio use inductor components (in addition to
resistors and capacitors) to achieve sharper or more narrow peaks of output vs.
frequency (improved selectivity). Electro-acoustic (quartz crystal) and other filter
devices are also used.
• Other circuit configurations may be used to produce “high pass”
and “low pass” filters.
– Low pass filters are used to attenuate the higher audio frequency
components of a voice waveform before digital sampling and coding.
Rev.5;Page. 37
©1996-2005 R.Levine
Radio Tuning Filter Example
C
R
I
L
+
V
-
~
~
~
|V/I|
“Band-pass” Graphic
Filter Symbol
Bandwidth at half
Power points is
R/(4••L).
70.7% voltage point
is the 50% power point.
fo=1/(2••L•C)
• Filter circuit using both inductors and capacitors
produces a “sharper” peak response than resistorcapacitor filters
– It can “filter” one desired radio frequency waveform out of many
different frequencies (attenuates other frequency ranges)
– Small R value implies very narrow bandwidth, good “selectivity”
• Filter center frequency fo is modified by mechanically
adjustable capacitor value C in older radios.
– Newer radio receivers have a fixed fo center frequency, and a
variable frequency (“local”) oscillator. A replica of the desired
radio frequency is produced via an interaction between the
desired frequency waveform and the local oscillator waveform in
a non-linear circuit. This device is called a frequency-adjustable
down-convertor.
Rev.5;Page. 38
©1996-2005 R.Levine
f
Illustrating Fourier* Series
• The following pages illustrate how a square
wave can be approximated by adding several
sine waves of appropriate frequencies and
amplitudes. These frequencies must be each an
exact integral multiple of the frequency of the
square wave signal to be approximated
• The coefficients can be computed by Fourier
analysis
– The coefficient for the nth term is proportional to the area
(integral) of the product waveform of the square wave and
the particular sine wave at that frequency
– For a square wave, only the 1,3,5,… odd “harmonic”
frequencies have non-zero coefficients.
*Invented by Joseph B.J.Fourier (pronounced: four-yay),
19th c. French mathematician and explorer.
Rev.5;Page. 39
©1996-2005 R.Levine
Approximate Square Wave
using Sine Waves
1.5
1
0.5
d
0
0.5
1
1.5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
t
1.5
1
0.5
v1 ( t )
v3 ( t )
0
v5 ( t )
0.5
1
1.5
0
0.1
0.2
0.3
0.4
0.5
t
Rev.5;Page. 40
©1996-2005 R.Levine
0.6
0.7
0.8
0.9
1
Approx. Square Wave Using 3
Odd Harmonics
1.5
1
1.0
0.5
v( t )
0
0.5
1.0
1
1.5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
t
Proper amplitude of each “harmonic” sine wave was found from a product
integral formula (same as statistical cross correlation).
Rev.5;Page. 41
©1996-2005 R.Levine
Approx. Square Wave Using 9
Odd Harmonics
2
1
1.0
v( t ) 0
1.0
1
2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
t
;
Ak
4.
 .f
;
k
Rev.5;Page. 42
A k. sin 2.  . fk. t
v( t )
©1996-2005 R.Levine
k
How Approximate?- I
• First, the computer software program* used
to produce these graphs is prone to some
errors of interpolation
– Only 100 data points are plotted for each graph
– The waveforms have spurious wiggles which are artifacts
of the software
• Second, when only a limited number of
harmonics are used, there is a “bump” or
overshoot peak at each corner of the
waveform
– Called “Gibbs’ Phenomena” (J.W.Gibbs, 19th c. American
Mathematician)
– Width of Gibbs’ bumps approaches zero as number of
additive terms increases
* Mathcad 4.0
Rev.5;Page. 43
©1996-2005 R.Levine
How Approximate?- II
• “Rise time” of steep slope at corners is faster (shorter
rise time) when more high frequency harmonics are
included in the Fourier series
• We loosely describe any waveform with sharp corners
and brief rise time as having “large bandwidth”
• Conversely, when we remove high frequency
components from a waveform via filtering devices, we:
– reduce the bandwidth (reduce the power of the high frequency
components of a base band waveform)
– increase the rise time of waveform “corners”
– change the waveform to some extent
• One objective of “high fidelity” is to retain (not remove)
the high frequency power from music or speech
Rev.5;Page. 44
©1996-2005 R.Levine
Comments on Examples
• These square waves use only sine components
(no cosines) because they are “odd parity”
symmetric about t=0 point. Even-symmetric
waveforms use only cosines. General nonsymmetric waveforms use both.
• Square waves have only odd harmonics
(1,3,5…) because they have no “wiggles” that
repeat 2, 4, 6… times in a cycle.
• General waveforms of arbitrary shape have
both odd and even harmonics, sine and cosine
components.
Rev.5;Page. 45
©1996-2005 R.Levine
“Musical” Analysis
•
•
•
•
Some* electronic musical instruments, like the original Hammond Organ, produce
complicated waveforms by combining higher frequency sine waves of
“harmonic” frequencies for each key on the musical keyboard.
These instruments use variable resistors to adjust the coefficients of each
“harmonic” component. The phase is not adjustable. Phase adjustment is not
needed for music, for reasons to be described.
Some combinations of harmonics are preset to approximate desired sounds like
“clarinet,” “trumpet,” etc.
Considering middle C (approx. 260 Hz**) as the fundamental frequency, the first
few “harmonics” are:
Frequency
260 Hz
2260=520 Hz
3260=780Hz
4260=1040Hz
5260=1300Hz
•
Musical Notation
Middle C
C (one octave) above Middle C (or C’)
G above C above Middle C, or G’
C above C above Middle C, or C’’
E above C above C above Mid.C, or E’’
If we “shift” all of these into the same octave, the result is a major triad chord (C,
E, G, C)!
*Other modern music synthesizers store the music time waveform using digital samples, similar to the samples
in a digital telephone system.
*More precisely, middle C is 261.1725 Hz in modern even-tempered concert pitch intonation (with A=440 Hz
exactly).
Rev.5;Page. 46
©1996-2005 R.Levine
“Peak Clipping” makes all
waveforms “square-ish”
• A periodic waveform that is modified by a non-linear
system so that the positive and negative peaks of each
cycle are clipped or flattened, will have a waveform
somewhat like a square wave
• The square wave comprises numerous odd harmonic
sine wave components
– One method for quantitatively describing the amount of
distortion in “high fidelity” amplifiers is to measure the amount
of undesired 3rd harmonic frequency power when a “pure” sine
wave is used as input.
– This is expressed as a logarithmic power ratio (in decibels)
compared to the fundamental frequency signal power.
• This is only one example of how a non-linear system
produces outputs at different sine wave frequencies
than the input frequency
Rev.5;Page. 47
©1996-2005 R.Levine
More Fun with Fourier Analysis
Other aspects:
•
Certain analysis and design processes, done mentally or with paper and pencil, are
much simpler and faster to do using sine wave components and Fourier analysis. This
was particularly significant before the days of widely available computers.
•
Time derivative and time integral of a waveform can be computed readily from the Fourier series or
transform, by multiplication or division by the frequency value (where j represents the “imaginary”
square root of negative 1, also called “i” in math and physics documents)
(d/dt)  h(t) <--->
h(t)dt <--->
j2fH(f)
H(f) / (j2f)
• Waveform filtering to remove undesired frequency
components
• Analysis of speech and other waveforms for efficient coding
• Discrete Cosine Transforms for image analysis and coding
– Function to be analyzed into component sine waves is a graph of color
brightness as a function of x-axis distance measured across the picture
or image
Rev.5;Page. 48
©1996-2005 R.Levine
Partial Spectrum Information is
Adequate for Voice Signals
• In general, we need both amplitude and phase (as a function
of frequency) to describe or characterize a device such as an
amplifier or an analog transmission system
• Data is usually expressed as two graphs
» amplitude vs. frequency
» phase vs. frequency (or time delay vs. frequency)
– Sometimes confusingly plotted on same graph paper using two different
vertical axis labels
• For voice, music or other signals destined for human ears,
experiments show that the phase is not significant, so it is
traditionally omitted. Altering the phase of various sine wave
components alters the resulting composite waveform, but it
sounds the same to human listeners.
• For MODEMs or other waveform-related devices which require
accurate phase reproduction, phase data cannot be omitted
Rev.5;Page. 49
©1996-2005 R.Levine
Decibels and Other Logarithmic
Units
• The ratio of two power values or levels is traditionally
expressed by a special logarithmic unit in the
telecommunication industry
• The ratio of two power levels (for example the input
and output of an amplifier, or a length of transmission
line) is expressed logarithmically:
10•log10 (P2/P1)
• or 20•log10 (V2/V1) when both voltages are measured in
parts of the system where the V/I ratio (the
transmission impedance) is the same.
Rev.5;Page. 50
©1996-2005 R.Levine
Memorable dB Values
Power Ratio
1000/1
100/1
10/1
4/1
2/1
1/1
1/2
1/10
1/100
Volt* Ratio
dB
31.62/ 1
10./1
3.16/1
2./1
1.41/1
1./1
0.707/1
0.316/1
0.1/1
+30.
+20.
+10.
+6.
+3. (more precisely 3.0103)
0.
-3.
-10.
-20.
* Power (the product of voltage•current) is proportional to the
square of voltage, when both measurements in the ratio are
made so that the ratio of voltage/current (the “impedance”) is
the same in the device being measured.
Rev.5;Page. 51
©1996-2005 R.Levine
dB Reference Levels
• The dB is the logarithm of a ratio of two quantities, not
an absolute unit
• In the telecommunications industry, 1 mW (1 milliwatt;
0.001 watt) is the usual reference level for most
electrical power measurements. Measurements with
this reference are designated dBm.
– Used also for radio receiver power measurements
– Used also for some optical fiber power measurement
• For sound intensity measurement in air, a power
intensity of 10-12 W/meter2 is the reference level. This is
not the same as 1 mW, and in fact they refer to different
physical quantities.
Rev.5;Page. 52
©1996-2005 R.Levine
Historical Information
• The “unit” for a decibel ratio is named for A.G. Bell, and
originated ~1910 in AT&T.
• One dB corresponds closely to the input/output power
ratio for one mile (1.6 km) of 19 gauge (0.91 mm diam.)
copper wire loop using a test signal at 1 kHz frequency
(today we mostly use thinner telephone wire)
• Before ~1910 the fractional power loss of 1 mile of such wire was
confusingly called a “mile”
• Nobody uses “whole” bels as units. A bel is expressed as
10 dB.
• Note the use of capital B, small d
– General rule for a scientific unit based on a person’s name:
capitalize the first abbreviation letter only, not in the full name. Thus
hertz (=cycles/second) or Hz. The capital H is used on the full word
only when describing the person (Heinrich Hertz) and not the
frequency unit hertz.
Rev.5;Page. 53
©1996-2005 R.Levine