Chapter 8
• Sine wave
• Fourier series
• Fourier transform
09/16/2010
© 2010 NTUST
Wave
• A wave is a disturbance. Unlike water waves, electrical waves
cannot be seen directly but they have similar characteristics.
All periodic waves can be constructed from sine waves, which
is why sine waves are fundamental.
Sine Waves
• The sinusoidal waveform (sine wave) is the fundamental
alternating current (ac) and alternating voltage waveform.
• Electrical sine waves
are named from the
mathematical function
with the same shape.
Period of a Sine Wave
Sine Waves
• Sine waves are characterized by the amplitude and period. The
amplitude is the maximum value of a voltage or current; the
period is the time interval for one complete cycle.
20 V
15 V
The amplitude (A)
of this sine wave
is 20 V
The period is 50.0 s
A
10 V
0V
t (s)
25
0
-10 V
-15 V
-20 V
T
37.5
50.0
Sine Waves
• The period of a sine wave can be measured between any
two corresponding points on the waveform.
TT T T
A
T
T
• By contrast, the amplitude of a sine wave is only measured
from the center to the maximum point.
Frequency
• Frequency ( f ) is the number of cycles that a sine wave
completes in one second.
• Frequency is measured in hertz (Hz).
If 3 cycles of a wave occur in one second, the frequency
is 3.0 Hz
1.0 s
Frequency of a Sine Wave
Period and Frequency
• The period and frequency are reciprocals of each other.
1
f 
T
and
1
T 
f
• Thus, if you know one, you can easily find the other.
(The 1/x key on your calculator is handy for converting between f and T.)
If the period is 50 s, the frequency is
0.02 MHz = 20 kHz.
Generation of a Sine Wave
• Sinusoidal voltages are produced by ac generators and
electronic oscillators.
• When a conductor rotates in a constant magnetic field, a
sinusoidal wave is generated.
C
N
D
B
S
A
B
C
D
A
Motion of conductor
Conduc tor
When
is moving parallel
When the
loopthe
is conductor
moving perpendicular
to the
of flux, voltage
no voltage
is induced.
lines of with
flux,the
thelines
maximum
is induced.
AC Generator (Alternator)
• Generators convert rotational energy to electrical energy. A
stationary field alternator with a rotating armature is shown.
The armature has an induced voltage, which is connected
through slip rings and brushes to a load. The armature loops
are wound on a magnetic core (not shown for simplicity).
• Small alternators may use a
permanent magnet as shown
here; other use field coils to
produce the magnetic flux.
N
brushes
arm ature
slip rings
S
AC Generator (Alternator)
• By increasing the number of poles, the number of cycles per
revolution is increased. A four-pole generator will produce two
complete cycles in each revolution.
Function Generator
Readout
Typical controls:
Function selection
Frequency
Range
Adjust
Outputs
Output level (amplitude)
DC offset
Sine
Duty cycle
CMOS output
Square
Triangle
Sine Wave Voltage and Current
• There are several ways to specify the voltage of a sinusoidal
voltage waveform. The amplitude of a sine wave is also
called the peak value, abbreviated as VP for a voltage
waveform.
20 V
15 V
VP
10 V
The peak voltage of
this waveform is 20 V.
0V
-10 V
-15 V
-20 V
t (s)
0
25
37.5
50.0
Sine Wave Voltage and Current
• The voltage of a sine wave can also be specified as either the
peak-to-peak or the rms value. The peak-to-peak is twice the
peak value. The rms value is 0.707 times the peak value.
20 V
15 V
The peak-to-peak
voltage is 40 V.
The rms voltage
is 14.1 V.
10 V
Vrms
0V
-10 V
-15 V
-20 V
0
VPP
t (s)
25
37.5
50.0
Sine Wave Voltage and Current
• For some purposes, the average value (actually the halfwave average) is used to specify the voltage or current. By
definition, the average value is as 0.637 times the peak
value.
20 V
The average value for
the sinusoidal voltage
is 12.7 V.
15 V
10 V
Vavg
0V
-10 V
-15 V
-20 V
t (s)
0
25
37.5
50.0
Sine Wave Voltage and Current
Sine Wave Voltage and Current
Sine Wave Voltage and Current
Sine Wave Voltage and Current
Sine Wave Voltage and Current
Sine Wave Voltage and Current
Angular Measurement
• Angular measurements can be made in degrees (o) or radians.
The radian (rad) is the angle that is formed when the arc is
equal to the radius of a circle. There are 360o or 2p radians
in one complete revolution.
1.0
0.8
R
0.6
0.4
0.2
R
0
-0.2
-0.4
-0.6
-0.8
-1.0
0
p
p
4
2
3p
4
p
5p
4
3p
2
7p
4
2p
Angular Measurement
Angular Measurement
Angular Measurement
• Because there are 2p radians in one complete revolution and
360o in a revolution, the conversion between radians and
degrees is easy to write. To find the number of radians, given
the number of degrees:
rad 
2p rad
 degrees
360
• To find the number of degrees, given the radians:
deg 
360
 rad
2p rad
Sine Wave Equation
Sine Wave Equation
• Instantaneous values of a wave are shown as v or i. The
equation for the instantaneous voltage (v) of a sine wave is
v  V p sin 
where
Vp = Peak voltage
 = Angle in rad or degrees
If the peak voltage is 25 V, the instantaneous
voltage at 50 degrees is 19.2 V
Sine Wave Equation
Examples
Examples
Sine Wave Equation
• A plot of the example in the previous slide (peak at 25 V)
is shown. The instantaneous voltage at 50o is 19.2 V as
previously calculated.
90
Vp
Vp = 25 V
v = Vp sin = 19.2 V
= 50
0
50
Vp
Examples
Examples
Phasor
• The sine wave can be represented as the projection of a
vector rotating at a constant rate. This rotating vector is
called a phasor. Phasors are useful for showing the phase
relationships in ac circuits.
90
180
0
0
90
180
360
Phase Shift
• The phase of a sine wave is an angular measurement that
specifies the position of a sine wave relative to a reference.
To show that a sine wave is shifted to the left or right of
this reference, a term is added to the equation given
previously.
v  VP sin   f 
where
f = Phase shift
Phase Shift
Example of a wave that lags the reference
…and the equation
has a negative
phase shift
Referenc e
40
Peak voltage
30
v = 30 V sin ( - 45o)
Voltage (V)
20
10
0
0
45
90
135 180
225
270
315
-20
-30
- 40
Notice that a lagging sine
wave is below the axis at 0o
Angle ()
360
405
Phase Shift
Example of a wave that leads the reference
Notice that a leading sine
wave is above the axis at
o
0eak
P
voltage
Referenc e
40
30
Voltage (V)
20
v = 30 V sin ( + 45o)
10
-45
0 0
-10
-20
-30
-40
45
90 135
180
225
…and the equation
has a positive
phase shift
Angle ()
270
315
360
Power in Resistive AC Circuits
• The power relationships developed for dc circuits apply to ac
circuits except you must use rms values when calculating
power. The general power formulas are:
P  Vrms I rms
2
Vrms
P
R
2
P  I rms
R
Power in Resistive AC Circuits
Assume a sine wave with a peak value of 40 V is
applied to a 100 W resistive load. What power is
dissipated?
40
30
Voltage (V)
20
10
0
-1 0
-2 0
-3 0
- 40
Vrms = 0.707 x Vp = 0.707 x 40 V = 28.3 V
2
Vrms
28.3 V 2
P

 8W
R
100 W
Instantaneous Value
Superimposed DC and AC Voltage
• Frequently dc and ac voltages are together in a waveform.
They can be added algebraically, to produce a composite
waveform of an ac voltage “riding” on a dc level.
Superimposed DC and AC Voltage
Examples
Examples
Examples
Examples
Pulse Definitions
Ideal pulses
Leading (rising) edge
Leading (falling) edge
Trailing (falling) edge
Trailing (rising) edge
Baseline
Am plitude
Am plitude
Baseline
Pulse
width
(a) Positive-going pulse
Pulse
width
(b) Negative-going pulse
Pulse Definitions
Non-ideal pulses
A
0.9 A
A
0.5 A
0.1A
t
tr
(a) Rise and fall times
t
tW
tf
(b) Pulse width
Notice that rise and fall times are measured
between the 10% and 90% levels whereas pulse
width is measured at the 50% level.
Repetitive Pulses
Examples
Examples
Triangular and Sawtooth Wave
• Triangular and sawtooth waveforms are formed by voltage
or current ramps (linear increase/decrease)
• Triangular waveforms
have positive-going
and negative-going
ramps of equal
duration.
• The sawtooth waveform
consists of two ramps,
one of much longer
duration than the other.
Harmonics
• All repetitive non-sinusoidal waveforms are composed of a
fundamental frequency (repetition rate of the waveform) and
harmonic frequencies.
• Odd harmonics are frequencies that are odd multiples of the
fundamental frequency.
• Even harmonics are frequencies that are even multiples of
the fundamental frequency.
Harmonics
• A square wave is composed only of the fundamental
frequency and odd harmonics (of the proper amplitude).
Oscilloscope
Display
Thesection
oscilloscope
is divided into
four main sections.
VerticalVertical
sectionsection
Signal coupling
Signal coupling
AC
AC
Ch 1
Ch 1
v
Volts/Di Volts/Di
v
DC
DC
GND
Display section
Amp
GND
Amp
From
Analog
Conversion/storage vertic al only
Conversion/storage
(Digital scopes only) sec tion
(Digital scopes only)
Analog Intensity
only
Vertical
AC
Ch 2
Ch 2
AC
DC
DC
GND
Verticalposition
position Amp
GND
Amp
Intensity
Conversion/storage
(Digital scopes only)
Conversion/storage
(Digital scopes only)
From horizontal sec tion
Digital
Digital
only
only
Trigger section
Horizontal
Horizontal
section
Trigger sectionTrigger
External trigger
coupling
External trigger
coupling
External
trigger
source
AC
External
trigger
DC
AC
Trigger
levelTrigger
and
slopelevel and
Trigger
source
Ch 1
DC
Ch 2
Ch 1
Ext
Line
Ext
Line
slope
Ch 2
Trigger
Trigger
circuits
circuits
section
Control and process
(Digital
scopes
Control
andonly)
proc ess
Sec /Div
(Digital scopes only)
Time base
Sec/Di
v
Horizontal
Time base
position
Horizontal
position
AC
AC
Power supply
To display sec tion
DC to all sec tions
Oscilloscope
Vertical section
Signal coupling
Volts/Di v
AC
DC
Ch 1
GND
AC
Ch 2
DC
GND
Display section
Amp
Conversion/storage
(Digital scopes only)
Vertical
position
Amp
Analog
only
Intensity
Conversion/storage
(Digital scopes only)
Digital
only
Horizontal
section
Trigger section
External trigger
coupling
External
trigger
Trigger
source
AC
DC
Ch 1
Ext
Line
Trigger
level and
slope
Ch 2
Control and process
(Digital scopes only)
Sec /Div
Trigger
circuits
Time base
Horizontal
position
AC
Power supply
DC to all sec tions
© Copyright 2007 Prentice-Hall
Oscilloscopes
Display
Vertical Horizontal
Trigger
VERT
ICAL
VERT
ICAL
HORIZONT
AL
HORIZONT
AL
CH
HH
CH11 CH
CH22 BOT
BOT
TR
TIGGER
RIGGER
SLOPE
SLOPE
ÐÐ
POSIT
ION
POSIT
ION
POSIT
ION
POSIT
ION
VOLT
S/DIV
VOLT
S/DIV
VOLT
S/DIV
VOLT
S/DIV
++
POSIT
ION
POSIT
ION
LEVEL
LEVEL
SEC/DIV
SEC/DIV
SOUR
CE
SOUR
CE
CH
11
CH
CH
22
CH
55VV
22mmVV
55VV
22mmVV
COUPLING
COUPLING
COUPLING
COUPLING
AC-DC-GND
AC-DC-GND
AC-DC-GND
AC-DC-GND
5 5s s
5 5nsns
EXT
EXT
LINE
LINE
TR
TIG
RIGCOUP
COUP
DC
DC
DISPLAY
DISPLAY
PP
RR
OB
EECOMP
OB
COMP
55VV
INT
INTENSIT
ENSITYY
CH
CH11
CH
CH22
AC
AC
EXT
EXTTRIG
TRIG
Selected Key Terms
Sine wave A type of waveform that follows a cyclic
sinusoidal pattern defined by the formula y =
A sin .
Alternating current Current that reverses direction in response to a
change in source voltage polarity.
Period (T) The time interval for one complete cycle of a
periodic waveform.
Frequency (f) A measure of the rate of change of a periodic function;
the number of cycles completed in 1 s.
Hertz The unit of frequency. One hertz equals one cycle
per second.
Selected Key Terms
Instantaneous value The voltage or current value of a waveform at a
given instant in time.
Peak value The voltage or current value of a waveform at its
maximum positive or negative points.
Peak-to-peak value The voltage or current value of a waveform
measured from its minimum to its maximum points.
rms value The value of a sinusoidal voltage that indicates its
heating effect, also known as effective value. It is
equal to 0.707 times the peak value. rms stands for
root mean square.
Selected Key Terms
Radian A unit of angular measurement. There are 2p
radians in one complete 360o revolution.
Phase The relative angular displacement of a time-varying
waveform in terms of its occurrence with respect to a
reference.
Amplitude The maximum value of a voltage or current.
Pulse A type of waveform that consists of two equal and
opposite steps in voltage or current separated by a time
interval.
Harmonics The frequencies contained in a composite waveform,
which are integer multiples of the pulse repetition
frequency.
Quiz
1. In North America, the frequency of ac utility voltage
is 60 Hz. The period is
a. 8.3 ms
b. 16.7 ms
c. 60 ms
d. 60 s
Quiz
2. The amplitude of a sine wave is measured
a. at the maximum point
b. between the minimum and maximum points
c. at the midpoint
d. anywhere on the wave
Quiz
3. An example of an equation for a waveform that lags
the reference is
a. v = -40 V sin ()
b. v = 100 V sin ( + 35o)
c. v = 5.0 V sin ( - 27o)
d. v = 27 V
Quiz
4. In the equation v = Vp sin  , the letter v stands for
the
a. peak value
b. average value
c. rms value
d. instantaneous value
Quiz
5. The time base of an oscilloscope is determined by
the setting of the
a. vertical controls
b. horizontal controls
c. trigger controls
d. none of the above
Quiz
6. A sawtooth waveform has
a. equal positive and negative going ramps
b. two ramps - one much longer than the other
c. two equal pulses
d. two unequal pulses
Quiz
7. The number of radians in 90o are
a. p/2
b. p
c. 2p/3
d. 2p
Quiz
8. For the waveform shown, the same power would be
delivered to a load with a dc voltage of
a. 21.2 V
60 V
45 V
b. 37.8 V
30 V
c. 42.4 V
0V
d. 60.0 V
-30 V
-45 V
-60 V
t (s)
0
25
37.5
50.0
Quiz
9. A square wave consists of
a. the fundamental and odd harmonics
b. the fundamental and even harmonics
c. the fundamental and all harmonics
d. only the fundamental
Quiz
10. A control on the oscilloscope that is used to set the
desired number of cycles of a wave on the display is
a. volts per division control
b. time per division control
c. trigger level control
d. horizontal position control
Quiz
Answers:
1. b
6. b
2. a
7. a
3. c
8. c
4. d
9. a
5. b
10. b
Fourier Series
Jean Baptiste Joseph Fourier
(French)(1763~1830)
Fourier Series
• 任一週期(periodic)函數可以分解成許多不
同振幅(amplitude)，不同頻率(frequency)
的
– 正弦(sinusoidal)諧波(harmonic)
與
– 餘弦(cosinusoidal)諧波(harmonic)的合成
(composition)
– A harmonic of a wave is a
component frequency of the signal that is
an integer multiple of the fundamental frequency,
i.e. if the fundamental frequency is f, the
harmonics have frequencies 2f, 3f, 4f, . . . etc.
Fourier Series
• A function f(x) can be expressed as a series of sines and
cosines:
• where:
Square Wave
Adding Harmonics
Three Harmonics
Combination of Three Harmonics
Square Wave
• Any periodic function can be expressed as the sum of a series
of sines and cosines (of varying amplitudes)
頻譜比較
Sawtooth Wave
Fourier Series
• 尤拉公式: establishes the deep
relationship between the
trigonometric functions and
the complex exponential
function. Euler's formula states
that, for any real number φ,
eiφ = cosφ + isinφ
其概念與複數平面之極式相通
Fourier Series
• 以複數型式表示傅立葉級數，將更為簡潔
Discrete Fourier Transform (DFT)
• 在處理信號時，常藉由離散傅立葉轉換(Discrete Fourier
Transform, DFT)來取得信號所對應的頻譜；再由頻譜來
讀取信號的參數。
• 但由於離散傅立葉所做的計算量過於龐大，當處理大量
的資料時，需要快速計算的演算法。
Discrete Fourier Transform (DFT)
• 以數位方式對連續信號取樣，週期時間T之內，可取樣N
個取樣點的數位信號
• DFT 可表為
式中m為頻域上的第m個刻度，n為時域上的第n個刻度
• X(m)為頻域上第m個刻度向量，x(n)為時域上第n個刻度
純量
Discrete Fourier Transform
• Forward DFT:
• Inverse DFT:
The complex numbers f0 … fN
are transformed into complex
numbers F0 … Fn
The complex numbers F0 … Fn
are transformed into complex
numbers
f0 … f N
DFT Example
• Interpreting a DFT can be slightly
difficult, because the DFT of real data
includes complex numbers.
• Basically:
– The magnitude of the complex number for
a DFT component is the power at that
frequency.
– The phase θ of the waveform can be
determined from the relative values of the
real and imaginary coefficients.
• Also both positive and “negative”
frequencies show up.
DFT Example
DFT Examples
DFT Examples
Fast Fourier Transform
• Discrete Fourier Transform would normally require O(n2)
time to process for n samples:
• Don’t usually calculate it this way in practice.
– Fast Fourier Transform takes O(n log(n)) time.
– Most common algorithm is the Cooley-Tukey Algorithm.
Fast Fourier Transform
• FFT (Fast Fourier Transform)，大幅提高頻譜的計算速度
• FFT使用條件：
– 信號必須是週期性的。
– 取樣週期必須為信號週期的整數倍。
– 取樣速率(Sampling rate)必須高於信號最高頻率的2 倍以上。
– 取樣點數N 必須為2k個資料。
快速傅利葉轉換原理
• A complex nth root of unity is a
complex number z such that zn = 1.
– n = e 2p i / n = principal n th root of
unity.
2 = i


3
1
4 = – e i t = cos t + i sin t.
1
– i2 = -1.
– There are exactly n roots of
5

unity: nk, k = 0, 1, . . . , n-1.
• n2= n/2
• nn+k= nk
0 = 1

6
=i
7
Fourier Cosine Transform
• Any function can be split into even and odd parts:
• Then the Fourier Transform can be re-expressed as:
Discrete Cosine Transform (DCT)
• When the input data contains only real numbers from an even
function, the sin component of the DFT is 0, and theDFT
becomes a Discrete Cosine Transform (DCT)
• There are 8 variants however, of which 4 are common.
DCT Types
• DCT Type II
– Used in JPEG, repeated for a 2-D transform.
– Most common DCT.
DCT Types
• DCT Type IV
– Used in MP3.
– In MP3, the data is overlapped so that half the data from one sample
set is reused in the next.
• Known as Modified DCT or MDCT
• This reduces boundary effects.
Why do we use DCT for Multimedia?
• For audio:
– Human ear has different dynamic range for different frequencies.
– Transform to from time domain to frequency domain, and quantize
different frequencies differently.
• For images and video:
– Human eye is less sensitive to fine detail.
– Transform from spacial domain to frequency domain, and quantize
high frequencies more coarsely (or not at all)
– Has the effect of slightly blurring the image - may not be perceptible
if done right.
Why use DCT/DFT?
• Some tasks are much easier to handle in the frequency
domain that in the time domain.
• Eg: graphic equalizer. We want to boost the bass:
1.
2.
3.
Transform to frequency domain.
Increase the magnitude of low frequency components.
Transform back to time domain.
Transformation
• Transformation from one domain to another
–
–
–
–
Fourier transform
Laplacian transform
Wavelet transform
Spherical harmonics transform
• Linear combination of a set of functions
– Basis functions
– Coefficient
Fourier Transform
• Fourier Series can be generalized to complex numbers, and
further generalized to derive the Fourier Transform.
Forward Fourier Transform:
Inverse Fourier Transform:
Note:
Fourier Transform
• Fourier Transform maps a time series (eg audio samples) into
the series of frequencies (their amplitudes and phases) that
composed the time series.
• Inverse Fourier Transform maps the series of frequencies
(their amplitudes and phases) back into the corresponding
time series.
• The two functions are inverses of each other.
.
Basic Properties
•
•
•
•
Linearity: h(x) = aƒ(x) + bg(x) 
Translation: h(x) = ƒ(x − x0) 
Modulation: h(x) = e2πixξ0ƒ(x) 
Scaling: h(x) = ƒ(ax) 
• Conjugation:
• Duality:
• Convolution:



Derivative Properties
F  f t   F   


f t e - jt dt
-
F -1F    f t  

j t


F

e
d

-

df (t ) dF -1F  
f ' t  


dt
dt



-

d  F  e jt d
-
dt
jF  e jt d   G ( )e jt d
-

de jt
  F  d
dt
-
where G ( )  jF    F  f ' (t )
.
Basic Properties
•
•
•
•
Linearity: h(x) = aƒ(x) + bg(x) 
Translation: h(x) = ƒ(x − x0) 
Modulation: h(x) = e2πixξ0ƒ(x) 
Scaling: h(x) = ƒ(ax) 
• Conjugation:
• Duality:
• Convolution:


