1. The document provides an overview of Fourier analysis techniques including Fourier series, Fourier transforms, and their applications to signal representation and analysis.
2. Key concepts covered include representing periodic and aperiodic signals in the time and frequency domains, properties of linear and time-invariant systems, Parseval's theorem relating signal energy in the time and frequency domains, and the Fourier transforms of basic functions like impulses and complex exponentials.
3. The document establishes essential mathematical foundations for further study of analog and digital communications techniques that involve signal processing and transmission in the frequency domain.
This presentation describes the Fourier Transform used in different mathematical and physical applications.
The presentation is at an Undergraduate in Science (math, physics, engineering) level.
Please send comments and suggestions to improvements to solo.hermelin@gmail.com.
More presentations can be found at my website http://www.solohermelin.com.
This presentation describes the Fourier Transform used in different mathematical and physical applications.
The presentation is at an Undergraduate in Science (math, physics, engineering) level.
Please send comments and suggestions to improvements to solo.hermelin@gmail.com.
More presentations can be found at my website http://www.solohermelin.com.
Presentation by Pipso Asiala of the activities and projects of the Media Lab department at Aalto University, Helsinki, Finland, made during the Cumulus Digital Culture working group session in Denver, Colorado, Sep. 29 2011.
Innovation at OnTheWight - Presented at What's next for Community Journalism ...onthewight
The presentation Simon gave at the "What's next for Community Journalism" conference organised by Centre for Community Journalism in Cardiff in September 2015.
We were asked to give a talk with an overview of innovations that we'd carried out over our ten years of doing #hyperlocal news on the Isle of Wight.
As the early innovation was so long ago, much of it has become mainstream in the intervening years!
Find the compact trigonometric Fourier series for the periodic signal.pdfarihantelectronics
Find the compact trigonometric Fourier series for the periodic signal x(t) and sketch the
amplitude and phase spectrum for first 4 frequency components. By inspection of the spectra,
sketch the exponential Fourier spectra. By inspection of spectra in part b), write the exponential
Fourier series for x(t)
Solution
ECE 3640 Lecture 4 – Fourier series: expansions of periodic functions. Objective: To build upon
the ideas from the previous lecture to learn about Fourier series, which are series representations
of periodic functions. Periodic signals and representations From the last lecture we learned how
functions can be represented as a series of other functions: f(t) = Xn k=1 ckik(t). We discussed
how certain classes of things can be built using certain kinds of basis functions. In this lecture we
will consider specifically functions that are periodic, and basic functions which are
trigonometric. Then the series is said to be a Fourier series. A signal f(t) is said to be periodic
with period T0 if f(t) = f(t + T0) for all t. Diagram on board. Note that this must be an everlasting
signal. Also note that, if we know one period of the signal we can find the rest of it by periodic
extension. The integral over a single period of the function is denoted by Z T0 f(t)dt. When
integrating over one period of a periodic function, it does not matter when we start. Usually it is
convenient to start at the beginning of a period. The building block functions that can be used to
build up periodic functions are themselves periodic: we will use the set of sinusoids. If the period
of f(t) is T0, let 0 = 2/T0. The frequency 0 is said to be the fundamental frequency; the
fundamental frequency is related to the period of the function. Furthermore, let F0 = 1/T0. We
will represent the function f(t) using the set of sinusoids i0(t) = cos(0t) = 1 i1(t) = cos(0t + 1)
i2(t) = cos(20t + 2) . . . Then, f(t) = C0 + X n=1 Cn cos(n0t + n) The frequency n0 is said to be
the nth harmonic of 0. Note that for each basis function associated with f(t) there are actually two
parameters: the amplitude Cn and the phase n. It will often turn out to be more useful to
represent the function using both sines and cosines. Note that we can write Cn cos(n0t + n) = Cn
cos(n) cos(n0t) Cn sin(n)sin(n0t). ECE 3640: Lecture 4 – Fourier series: expansions of periodic
functions. 2 Now let an = Cn cos n bn = Cn sin n Then Cn cos(n0t + n) = an cos(n0t) + bn
sin(n0t) Then the series representation can be f(t) = C0 + X n=1 Cn cos(n0t + n) = a0 + X n=1 an
cos(n0t) + bn sin(n0t) The first of these is the compact trigonometric Fourier series. The second
is the trigonometric Fourier series.. To go from one to the other use C0 = a0 Cn = p a 2 n + b 2 n
n = tan1 (bn/an). To complete the representation we must be able to compute the coefficients.
But this is the same sort of thing we did before. If we can show that the set of functions
{cos(n0t),sin(n0t)} is in fact an orthogonal set, then we can use the same.
4. a Find the compact trigonometric Fourier series for the periodic s.pdfjovankarenhookeott88
4. a Find the compact trigonometric Fourier series for the periodic signal x() and sketch the
amplitude and phase spectrum for first 4 frequency components. By inspection of the spectra,
sketch the exponential Fourier spectra. for x(t) c) By inspection of spectra in part b), write the
exponential Fourier series (20 points) X(t) T/2 a/2 T/2
Solution
The Fourier series is a method of expressing most periodic, time-domain functions in the
frequency domain. The frequency domain representation appears graphically as a series of spikes
occurring at the fundamental frequency (determined by the period of the original function) and
its harmonics. The magnitudes of these spikes are the Fourier coefficients. This series of
components are called the wave spectrum. THE EXPONENTIAL FOURIER SERIES å ¥ =-¥ w
= n jn t n f t D e 0 ( ) where: D is the amplitude or coefficient n is the harmonic w0 is the radian
frequency [rad/s] w0/2p is the frequency [hertz] T0 = 2p/w0 is the period [sec.] ò - w = 0 0 ( ) 1
0 T jn t n f t e dt T D 0 0 0 D = C = a 0 0 2 T p w = The exponential form carries no more and no
less information than the other forms but is preferred because it requires less calculation to
determine and also involves simpler calculations in its use. One drawback is that the exponential
form is not as easy to visualize as trigonometric forms. THE TRIGONOMETRIC FOURIER
SERIES å ¥ = = + w + w 1 0 0 0 ( ) cos sin n n n f t a a n t b n t where: a is the amplitude or
coefficient n is the harmonic q is the phase w0 is the radian frequency [rad/s] w0/2p is the
frequency [hertz] T0 = 2p/w0 is the period [sec.] ò = 0 ( ) 1 0 0 T f t dt T a 0 0 2 T p w = n n T n
f t n t dt C T a = w = q ò ( ) cos cos 2 0 0 0 n n T n f t n t dt C T b = w = - q ò ( )sin sin 2 0 0 0
The Fourier series of an even periodic function will consist of cosine terms only and an odd
periodic function will consist of sine terms only.
Tom Penick tomzap@eden.com www.teicontrols.com/notes 2/5/2000 THE COMPACT
TRIGONOMETRIC FOURIER SERIES å ¥ = = + w + q 1 0 0 ( ) cos( ) n n n f t C C n t where:
C is the amplitude or coefficient n is the harmonic q is the phase w0 is the radian frequency
[rad/s] w0/2p is the frequency [hertz] T0 = 2p/w0 is the period [sec.] 2 2 Cn = an + bn 0 0 C = a
n n a , b : see above ÷ ÷ ø ö ç ç è æ - q = - n n n a 1 b tan 0 0 2 T p w = THE DIRICHLET
CONDITIONS WEAK DIRICHLET CONDITION - For the Fourier series to exist, the function
f(t) must be absolutely integrable over one period so that coefficients a0, an, and bn are finite.
This guarantees the existence of a Fourier series but the series may not converge at every point.
STRONG DIRICHLET CONDITIONS - For a convergent Fourier series, we must meet the
weak Dirichlet condition and f(t) must have only a finite number of maxima and minima in one
period. It is permissible to have a finite number of finite discontinuities in one period.
FREQUENCY SPECTRA AMPLITUDE SPECTRUM - The plot of amplitude Cn versus the
(radian) frequency.
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Synthetic Fiber Construction in lab .pptxPavel ( NSTU)
Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
Macroeconomics- Movie Location
This will be used as part of your Personal Professional Portfolio once graded.
Objective:
Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
Acetabularia Information For Class 9 .docxvaibhavrinwa19
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Palestine last event orientationfvgnh .pptxRaedMohamed3
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Francesca Gottschalk - How can education support child empowerment.pptxEduSkills OECD
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The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
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The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
2. Signal Representation
s(t) = A sin(2π fo t +φo ) or A sin(ωo t +φo )
Time-domain: waveform
A: Amplitude
Time (seconds) f : Frequency (Hz) (ω=2πf)
φ : Phase (radian or degrees)
Period (seconds)
S(f)
Frequency-domain: spectrum
fo Frequency (Hz)
p. 2
3. Energy and Power of Signals
For an arbitrary signal f(t), the total energy normalized to unit
resistance is defined as
∆ T
E = lim ∫ f (t ) 2 dt joules,
T →∞ −T
and the average power normalized to unit resistance is defined as
∆ 1 T
P = lim
T → ∞ 2T ∫
−T
f (t ) 2 dt watts ,
• Note: if 0 < E < ∞ (finite) P = 0.
• When will 0 < P < ∞ happen?
p. 3
4. Periodic Signal
A signal f(t) is periodic if and only if
f (t + T0 ) = f (t ) for all t (*)
where the constant T0 is the period.
The smallest value of T0 such that equation (*) is satisfied is
referred to as the fundamental period, and is hereafter simply
referred to as the period.
Any signal not satisfying equation (*) is called aperiodic.
p. 4
5. Deterministic & Random Signals
Deterministic signal can be modeled as a completely specified
function of time.
Example
f (t ) = A cos( ω 0 t + θ )
Random signal cannot be completely specified as a function of
time and must be modeled probabilistically.
p. 5
6. System
Mathematically, a system is a rule used for assigning a function g(t)
(the output) to a function f(t) (the input); that is,
g(t) = h{ f(t) }
where h{•} is the rule or we call the impulse response.
f(t) h(t) g(t)
For two systems connected in cascade, the output of the first system
forms the input to second, thus forming a new overall system:
g(t) = h2 { h1 [ f(t) ] } = h{ f(t) }
p. 6
7. Linear System
If a system is linear then superposition applies; that is, if
g1(t) = h{ f1(t) }, and g2(t) = h{ f2(t) }
then
h{ a1 f1(t) + a2 f2(t) } = a1 g1(t) + a2 g2(t) (*)
where a1, a2 are constants. A system is linear if it satisfies
Eq. (*); any system not meeting these requirement is nonlinear.
p. 7
8. Time-Invariant and Time-Varying
A system is time-invariant if a time shift in the input results
in a corresponding time shift in the output so that
g (t − t 0 ) = h{ f (t − t 0 )} for any t 0 .
The output of a time-invariant system depends on time differences and
not on absolute values of time.
Any system not meeting this requirement is said to be time-varying.
p. 8
9. Fourier Series
A periodic function of time s(t) with a fundamental period of T0 can be
represented as an infinite sum of sinusoidal waveforms. Such
summation, a Fourier series, may be written as:
∞
2 π nt ∞ 2 πnt
s (t ) = A0 + ∑ An cos + ∑ B n sin , (1)
n =1 T0 n =1 T0
where the average value of s(t), A0 is given by
1 T20
A0 =
T0 ∫− T20 s (t ) dt , (2)
while
2 T0
2 π nt
An = ∫ (3)
2
T0
s (t ) cos dt ,
T0 − 2 T0
and
2 T0
2 π nt
Bn = ∫
2
T0
s (t ) sin dt . (4)
T0 − 2 T0
p. 9
10. Fourier Series
An alternative form of representing the Fourier series is
∞
2 πnt
s (t ) = C 0 + ∑ C n cos
− φn
(5)
n =1 T0
where
C0 = A0 , (6)
2 2
Cn = An + B n , (7)
B
φ n = tan −1 n . (8)
An
The Fourier series of a periodic function is thus seen to consist of a
summation of harmonics of a fundamental frequency f0 = 1/T0.
The coefficients Cn are called spectral amplitudes, which represent the
amplitude of the spectral component Cn cos(2πnf0t − φn) at frequency
nf0.
p. 10
11. Fourier Series
The exponential form of the Fourier series is used extensively in
communication theory. This form is given by
∞ j 2 π nt
s (t ) = ∑S
n = −∞
n e T0
, (9)
where
1 T0
−
j 2 π nt
(10)
Sn = ∫ s (t ) e dt
2 T0
T0
T0 − 2
Note that Sn and S−n are complex conjugate of one another, that is
S n = S −n .
*
(11)
These are related to the Cn by
C n − jφ n (12)
S0 = C0 , Sn = e .
2
p. 11
12. Fourier Series
Amplitude Spectra (Line Spectra)
Fig.(a)
Cn
Note that except S0 = C0, each
0 fo 2fo 3fo 4fo 5fo 6fo (n-1) fo nfo spectral line in Fig. (a) at frequency f
is replaced by the two spectral lines in
Fig. (b), each with half amplitude,
Fig.(b) one at frequency f and one at
|Sn|
frequency - f.
••• •••
-nfo -(n-1)fo ••• - 6fo0-5fo -4fo -3fo -2fo -fo 0 fo 2fo 3fo 4fo 5fo 6fo ••• (n-1) fo nfo
p. 12
13. Fourier Series : Example
Consider a unitary square wave defined by The Bn coefficients are given by
1, 0 < t < 0.5 2 T0
2πnt
Bn = ∫
2
x(t ) = T0
x(t ) sin dt
T0 −2 T0
− 1, 0.5 < t < 1
= 2 ∫ x(t ) sin (2πnt )dt
1
and periodically extended outside this interval. 0
The average value is zero, so = 2 ∫ sin (2πnt )dt + 2 ∫ − sin (2πnt )dt
0.5 1
0 0.5
A0 = 0. cos(2πnt ) cos(2πnt )
1 0.5
= 2 − +
Recall that 2 T0
2πnt
2πn 0 2πn 0.5
An = ∫
2
x(t ) cos dt
T0
T0
−2 T0 2
= (1 − cos nπ)
πn
= 2 ∫ x(t ) cos(2πnt )dt
1
0
which results in
= 2 ∫ cos(2πnt )dt + 2 ∫ − cos(2πnt )dt
0.5 1
4
0 0.5
, n is odd
Bn = nπ
sin (2πnt ) sin (2πnt )
0.5 1
= 2 − 0,
n is even
2πn 0 2πn 0.5
=0
Thus all An coefficients are zero.
p. 13
14. Fourier Series : Example
The Fourier series of a square wave of unitary amplitude with odd symmetry is
therefore
4 1 1
x (t ) = (sin 2 πt + sin 6 πt + sin 10 πt + K)
π 3 5
1st term 1st + 2nd terms 1st + 2nd + 3rd terms
Sum up to the 6th term
p. 14
15. Fourier Transform
Representation of an Aperiodic Function
Consider an aperiodic function f(t)
To represent this function as a sum of exponential functions over
the entire interval (-∞, ∞), we construct a new periodic function
fT(t) with period T.
By letting T→∞,
lim f T (t ) = f (t ) (13)
T →∞
p. 15
16. Fourier Transform
The new function fT(t) can be represented by an exponential
Fourier series, which is written as
∞
f T (t ) = ∑ Fn e jn ω 0 t ,
n = −∞
(14)
where
1 T /2
(15)
Fn =
T ∫−T / 2
f T (t ) e − jn ω 0 t dt
and ω0 = 2π / T .
p. 16
17. Fourier Transform
For the sake of clear presentation, we set
∆ ∆
ω n = nω 0 , F ( ω n ) = TF n , (16)
Thus, Eq.(14) and (15) become
∞
1
f T (t ) = ∑T
n = −∞
F ( ω n ) e jω n t , (17)
T /2
(18)
F (ω n ) = ∫−T / 2
f T (t ) e − jω n t dt .
The spacing between adjacent lines in the line stream of fT(t)
is
∆ω = 2π / T . (19)
p. 17
18. Fourier Transform
Using this relation for T, we get
∞
∆ω
f T (t ) = ∑
n = −∞
F (ω n )e jω n t
2π
. (20)
As T becomes very large, ∆ω becomes smaller and the spectrum
becomes denser.
In the limit T → ∞, the discrete lines in the spectrum of fT(t) merge
and the frequency spectrum becomes continuous.
Therefore, 1 ∞
lim f T (t ) = lim
T →∞ T →∞ 2π
∑
n = −∞
F ( ω n ) e jω n t ∆ ω (21)
becomes 1 ∞
2 π ∫− ∞
f (t ) = F ( ω ) e jω t d ω (22)
p. 18
19. Fourier Transform
In a similar way, Eq. (18) becomes
∞
F (ω) = ∫−∞
f (t ) e − jω t dt . (23)
Eq. (22) and (23) are commonly referred to as the
Fourier transform pair.
Fourier Transform
∞
F (ω ) = ∫
−∞
f (t ) e − jω t dt
Inverse Fourier Transform
1 ∞
2 π ∫− ∞
f (t ) = F ( ω ) e jω t d ω
p. 19
20. Spectral Density Function
F(ω): The spectral density function of f(t).
Fig. 3.2
A unit gate function Its spectral density graph
sin( ω / 2 )
Sa ( ω / 2 ) =
ω/2
p. 20
21. Parseval’s Theorem
The energy delivered to a 1-ohm resistor is
∞ ∞
E= ∫ f (t ) dt = ∫ (24)
2
f (t ) f * (t ) dt .
−∞ −∞
Using Eq. (22) in (24), we get
∞ 1 ∞ * 1 ∞
E = ∫ f (t ) ∫ F (ω)e − jωt dω dt f (t ) = ∫− ∞ F (ω)e d ω
jω t
−∞
2π − ∞ 2π
1 ∞ * ∞
F (ω) ∫ f (t )e − jωt dt dω
2π ∫−∞
=
−∞
1 ∞ * (25)
=
2π ∫−∞ F (ω) F (ω)dω.
Parseval’s Theorem:
∞ 1 ∞
∫ 2 π ∫− ∞
2 2
−∞
f (t ) dt = F ( ω) d ω. (26)
p. 21
22. Fourier Transform: Impulse Function
The unit impulse function satisfies
∞
∫ δ( x)dx = 1, (27)
−∞
∞ x = 0,
δ ( x) = (28)
0 x ≠ 0.
Using the integral properties of the impulse function, the Fourier
transform of a unit impulse, δ(t), is
∞
ℑ{δ(t )} = ∫ δ(t )e − jωt dt = e j 0 = 1. (29)
−∞
If the impulse is time-shifted, we have
∞
ℑ{δ(t − t0 )} = ∫ δ(t − t0 )e − jωt dt = e − jωt0 . (30)
−∞
p. 22
23. Fourier Transform: Complex
Exponential Function
± jω t
The spectral density of e 0 will be concentrated at ±ω0.
1 ∞
ℑ {δ ( ω m ω 0 )} =
2 π ∫− ∞
−1
δ ( ω m ω 0 ) e jω t d ω
1 ± jω 0 t (31)
= e ,
2π
Taking the Fourier transform of both sides, we have
(32)
ℑℑ −1
2π
{
{δ ( ω m ω 0 ) } = 1 ℑ e ± j ω 0 t }
which gives
{ }
ℑ e ± j ω 0 t = 2πδ (ω m ω 0 ) (33)
p. 23
24. Fourier Transform: Sinusoidal Function
The sinusoidal signals cos ω0tand sin ωcan be written in terms of
0t
the complex exponentials.
Their Fourier transforms are given by
{
ℑ{cos ω 0 t } = ℑ 1 e jω 0 t + 1 e − jω 0 t
2 2
}
= πδ ( ω − ω 0 ) + πδ ( ω + ω 0 ),
(34)
ℑ{sin ω0t} = ℑ {1
2j e jω0t − 21j e − jω0t }
πδ(ω − ω0 ) − πδ(ω + ω0 )
= .
j
(35)
p. 24
25. Fourier Transform: Periodic Functions
We can express a function f(t) that is periodic with period T by its
exponential Fourier series
∞
f T (t ) = ∑ Fn e jn ω 0 t
n = −∞
where ω0 = 2π/T. (36)
Taking the Fourier transform, we have
∞ jnω0 t
ℑ{ fT (t )} = ℑ ∑ Fn e
e.g.
n = −∞
∑ F ℑ{e }
∞
jnω0t
= n A unit gate function Its Fourier transform
n = −∞
∞
= 2π ∑ Fn δ(ω − nω0 ).
n = −∞
(37) Line spectrum of f(t) Its spectral density graph
with period T p. 25
28. Properties of Fourier Transform
Linearity (Superposition) Time Shifting (Delay)
a1 f1 (t ) + a 2 f 2 (t ) ↔ a1 F1 ( ω ) + a 2 F2 ( ω ) f (t − t 0 ) ↔ F (ω ) e − jω t 0
Complex Conjugate Frequency Shifting (Modulation)
f * (t ) ↔ F * (−ω) f ( t ) e jω 0 t ↔ F ( ω − ω 0 )
Duality
Convolution
F (t ) ↔ 2 π f ( − ω ).
f1 (t ) ∗ f 2 (t ) ↔ F1 ( ω ) F2 ( ω )
Scaling
1 ω Multiplication
f (at ) ↔ F for a ≠ 0.
a a
f1 (t ) f 2 (t ) ↔ F1 ( ω ) ∗ F2 ( ω )
Differentiation
dn
f (t ) ↔ ( jω) n F (ω)
dt n
p. 28
29. Properties of Fourier Transform
Duality F (t ) ↔ 2 π f ( − ω).
Scaling 1 ω
f ( at ) ↔ F for a ≠ 0.
a a
p. 29
30. Properties of Fourier Transform
Frequency Shifting (Modulation)
jω 0 t
f (t ) e ↔ F (ω − ω 0 )
p. 30