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Fourier Series

When the French mathematician Joseph Fourier (1768–1830) was trying to solve a problem in heat conduction, he needed to express a function f as an inﬁnite series of sine and cosine functions:

1

f x

a0 n 1

a n cos nx a1 cos x b1 sin x

bn sin nx a3 cos 3x b3 sin 3x

a0

a2 cos 2x b2 sin 2x

Earlier, Daniel Bernoulli and Leonard Euler had used such series while investigating problems concerning vibrating strings and astronomy. The series in Equation 1 is called a trigonometric series or Fourier series and it turns out that expressing a function as a Fourier series is sometimes more advantageous than expanding it as a power series. In particular, astronomical phenomena are usually periodic, as are heartbeats, tides, and vibrating strings, so it makes sense to express them in terms of periodic functions. We start by assuming that the trigonometric series converges and has a continuous function f x as its sum on the interval , , that is,

2

f x

a0 n 1

a n cos nx

bn sin nx

x

Our aim is to ﬁnd formulas for the coefﬁcients a n and bn in terms of f . Recall that for a power series f x cn x a n we found a formula for the coefﬁcients in terms of derivn atives: cn f a n!. Here we use integrals. If we integrate both sides of Equation 2 and assume that it’s permissible to integrate the series term-by-term, we get

y

f x dx

y

a 0 dx

y n 1

a n cos nx

bn sin nx dx

2 a0 n 1

an y

cos nx dx n 1

bn y sin nx dx

But

y

cos nx dx

1 sin nx n sin nx dx f x dx

1 sin n n 0. So 2 a0

sin

n

0

because n is an integer. Similarly, x

y

1

2

❙❙❙❙

FOURIER SERIES

and solving for a0 gives

|||| Notice that a 0 is the average value of f over , . the interval

3

a0

1 2

y

f x dx

To determine an for n 1 we multiply both sides of Equation 2 by cos mx (where m is an integer and m 1) and integrate term-by-term from to :

y

4

f x cos mx dx

y n 1

a0 n 1

a n cos nx

bn sin nx

cos mx dx

a0 y

cos mx dx

an

y

cos nx cos mx dx n 1

bn

y

sin nx cos mx dx

We’ve seen that the ﬁrst integral is 0. With the help of Formulas 81, 80, and 64 in the Table of Integrals, it’s not hard to show that

y y

sin nx cos mx dx

0 0

for all n and m for n for n m m

cos nx cos mx dx

So the only nonzero term in (4) is am and we get

y

f x cos mx dx

am

Solving for am , and then replacing m by n, we have

5

an

1

y

f x cos nx dx

n

1, 2, 3, . . .

Similarly, if we multiply both sides of Equation 2 by sin mx and integrate from we get 1

to ,

6

bn

y

f x sin nx dx

n

1, 2, 3, . . .

We have derived Formulas 3, 5, and 6 assuming f is a continuous function such that Equation 2 holds and for which the term-by-term integration is legitimate. But we can still consider the Fourier series of a wider class of functions: A piecewise continuous function on a, b is continuous except perhaps for a ﬁnite number of removable or jump discontinuities. (In other words, the function has no inﬁnite discontinuities. See Section 2.5 for a discussion of the different types of discontinuities.)

FOURIER SERIES

❙❙❙❙

3

7 Definition Let f be a piecewise continuous function on Fourier series of f is the series

,

. Then the

a0 n 1

a n cos nx

bn sin nx

where the coefﬁcients an and bn in this series are deﬁned by a0 1 2

y

f x dx

an

1

y

f x cos nx dx

bn

1

y

f x sin nx dx

and are called the Fourier coefﬁcients of f . Notice in Deﬁnition 7 that we are not saying f x is equal to its Fourier series. Later we will discuss conditions under which that is actually true. For now we are just saying that associated with any piecewise continuous function f on , is a certain series called a Fourier series.

EXAMPLE 1 Find the Fourier coefﬁcients and Fourier series of the square-wave function

f deﬁned by f x 0 if 1 if 0 x x 0 and f x 2 f x

So f is periodic with period 2 and its graph is shown in Figure 1.

|||| Engineers use the square-wave function in describing forces acting on a mechanical system and electromotive forces in an electric circuit (when a switch is turned on and off repeatedly). Strictly speaking, the graph of f is as shown in Figure 1(a), but it’s often represented as in Figure 1(b), where you can see why it’s called a square wave. y 1

_π

0

π

2π

x

(a) y 1

_π

0

π

2π

x

FIGURE 1 Square-wave function

(b) SOLUTION Using the formulas for the Fourier coefﬁcients in Deﬁnition 7, we have

a0

1 2

y

f x dx

1 2

y

0

0 dx

1 2

y

0

1 dx

0

1 2

1 2

4

❙❙❙❙

FOURIER SERIES

and, for n

1, an 1

y

f x cos nx dx 1 sin nx n 1 n

1

y

0

0 dx

1

y

0

cos nx dx

0

sin n

sin 0

0

0

bn

1

y

f x sin nx dx

1

y

0

0 dx

1

y

0

sin x dx

1 cos nx n

|||| Note that cos n equals 1 if n is even and 1 if n is odd.

0

1 n

cos n

cos 0

0 2 n

if n is even if n is odd

The Fourier series of f is therefore a0 a1 cos x a2 cos 2x a3 cos 3x b3 sin 3x

b1 sin x 1 2 0 0 2 0 sin x

b2 sin 2x

0 sin 2x 2 sin 3x 3 2k

2 sin 3x 3 2 sin 5x 5

0 sin 4x 2 sin 7x 7

2 sin 5x 5

1 2

2

sin x

Since odd integers can be written as n Fourier series in sigma notation as 1 2

1, where k is an integer, we can write the

2 k 1

2k

1

sin 2k

1x

In Example 1 we found the Fourier series of the square-wave function, but we don’t know yet whether this function is equal to its Fourier series. Let’s investigate this question graphically. Figure 2 shows the graphs of some of the partial sums Sn x 1 2 2 sin x 2 sin 3x 3 2 sin nx n

when n is odd, together with the graph of the square-wave function.

FOURIER SERIES

❙❙❙❙

5

y 1 S¡ _π π x y 1 S£

y 1 S∞

_π

π

x

_π

π

x

y 1 S¶

y 1 S¡¡

y 1 S¡∞

_π

π

x

_π

π

x

_π

π

x

FIGURE 2 Partial sums of the Fourier series for the square-wave function

We see that, as n increases, Sn x becomes a better approximation to the square-wave function. It appears that the graph of Sn x is approaching the graph of f x , except where x 0 or x is an integer multiple of . In other words, it looks as if f is equal to the sum of its Fourier series except at the points where f is discontinuous. The following theorem, which we state without proof, says that this is typical of the Fourier series of piecewise continuous functions. Recall that a piecewise continuous function has only a ﬁnite number of jump discontinuities on , . At a number a where f has a jump discontinuity, the one-sided limits exist and we use the notation f a xla lim f x

f a

xla

lim f x

8 Fourier Convergence Theorem If f is a periodic function with period 2 and f and f are piecewise continuous on , , then the Fourier series (7) is convergent. The sum of the Fourier series is equal to f x at all numbers x where f is continuous. At the numbers x where f is discontinuous, the sum of the Fourier series is the average of the right and left limits, that is

1 2

f x

f x

If we apply the Fourier Convergence Theorem to the square-wave function f in Example 1, we get what we guessed from the graphs. Observe that f 0 xl0 lim f x

1

and

f 0

xl0

lim f x

0

and similarly for the other points at which f is discontinuous. The average of these left and right limits is 1 , so for any integer n the Fourier Convergence Theorem says that 2 1 2 2 k 1

2k

1

sin 2k

1x n .)

f x

1 2

if n if x

n n

(Of course, this equation is obvious for x

6

❙❙❙❙

FOURIER SERIES

Functions with Period 2L

If a function f has period other than 2 , we can ﬁnd its Fourier series by making a change f x for all x. If we let of variable. Suppose f x has period 2L, that is f x 2L t x L and tt f x f Lt L corresponds to t . The Fourier

then, as you can verify, t has period 2 and x series of t is a0 n 1

a n cos nt

bn sin nt

where a0 1 1 2

y

t t dt 1

an

y

t t cos nt dt Lt

bn

y

t t sin nt dt x L, dt L dx, and

If we now use the Substitution Rule with x we have the following

9

, then t

If f is a piecewise continuous function on a0 n 1

L, L , its Fourier series is bn sin n x L

a n cos

n x L 1 2L

where

|||| Notice that when L the same as those in (7). these formulas are

a0 and, for n an 1 L 1,

y

L L

f x dx

y

L

L

f x cos

n x L

dx

bn

1 L

y

L L

f x sin

n x L

dx

Of course, the Fourier Convergence Theorem (8) is also valid for functions with period 2L.

EXAMPLE 2 Find the Fourier series of the triangular wave function deﬁned by f x

x f x for all x. (The graph of f is shown in Figure 3.) for 1 x 1 and f x 2 For which values of x is f x equal to the sum of its Fourier series? y 1

FIGURE 3 The triangular wave function

_1

0

1

2

x

FOURIER SERIES

❙❙❙❙

7

SOLUTION We ﬁnd the Fourier coefﬁcients by putting L

1 2

1 in (9):

1 2

a0

|||| Notice that a 0 is more easily calculated as an area.

y

1 1

x dx

1 2

y

1 0

0 1 1 2

x dx

y

1

0

x dx

1 4

x2

]

0 1

1 4

x2

]

and for n

1, an

y

1 1

x cos n x dx

2 y x cos n x dx

0

1

because y x cos n x is an even function. Here we integrate by parts with u and dv cos n x dx. Thus, an x 2 sin n x n 0 2 n

1

x

0

2 n

1

y

1

0

sin n x dx 2 n2

cos n x n

2

cos n

1

0

Since y

x sin n x is an odd function, we see that bn

y

1 1

x sin n x dx

0

We could therefore write the series as 1 2 But cos n 2 cos n n2 1

2

cos n x

n 1

1 if n is even and cos n

1 if n is odd, so 0 if n is even 4 n2

2

an

2 n2

2

cos n

1

if n is odd

Therefore, the Fourier series is 1 2 4

2

cos

x

4 9 4

2

cos 3 x

4 25 1

2

cos 5 x

1 2

n 1

2k

1

2

2

cos 2k

x

The triangular wave function is continuous everywhere and so, according to the Fourier Convergence Theorem, we have f x 1 2 4 n 1

2k

1

2

2

cos 2k

1

x

for all x

8

❙❙❙❙

FOURIER SERIES

In particular, x 1 2 4 k 1

2k

1

2

2

cos 2k

1

x

for

1

x

1

Fourier Series and Music

One of the main uses of Fourier series is in solving some of the differential equations that arise in mathematical physics, such as the wave equation and the heat equation. (This is covered in more advanced courses.) Here we explain brieﬂy how Fourier series play a role in the analysis and synthesis of musical sounds. We hear a sound when our eardrums vibrate because of variations in air pressure. If a guitar string is plucked, or a bow is drawn across a violin string, or a piano string is struck, the string starts to vibrate. These vibrations are ampliﬁed and transmitted to the air. The resulting air pressure ﬂuctuations arrive at our eardrums and are converted into electrical impulses that are processed by the brain. How is it, then, that we can distinguish between a note of a given pitch produced by two different musical instruments? The graphs in Figure 4 show these ﬂuctuations (deviations from average air pressure) for a ﬂute and a violin playing the same sustained note D (294 vibrations per second) as functions of time. Such graphs are called waveforms and we see that the variations in air pressure are quite different from each other. In particular, the violin waveform is more complex than that of the ﬂute.

t

t

FIGURE 4 Waveforms

(a) Flute

(b) Violin

We gain insight into the differences between waveforms if we express them as sums of Fourier series: Pt a0 a1 cos t L b1 sin t L a2 cos 2 t L b2 sin 2 t L

In doing so, we are expressing the sound as a sum of simple pure sounds. The difference in sounds between two instruments can be attributed to the relative sizes of the Fourier coefﬁcients of the respective waveforms. The n th term of the Fourier series, that is, a n cos n t L bn n t L

is called the nth harmonic of P. The amplitude of the n th harmonic is An and its square, A2 n a2 n

sa 2 n

b2 n

b2 , is sometimes called energy of the n th harmonic. (Notice that n

FOURIER SERIES

❙❙❙❙

9

bn and for a Fourier series with only sine terms, as in Example 1, the amplitude is A n the energy is A2 b 2.) The graph of the sequence A2 is called the energy spectrum of n n n P and shows at a glance the relative sizes of the harmonics. Figure 5 shows the energy spectra for the ﬂute and violin waveforms in Figure 4. Notice that, for the ﬂute, A2 tends to diminish rapidly as n increases whereas, for the violin, the n higher harmonics are fairly strong. This accounts for the relative simplicity of the ﬂute waveform in Figure 4 and the fact that the ﬂute produces relatively pure sounds when compared with the more complex violin tones.

A@ n A@ n

FIGURE 5 Energy spectra

0

2

4

6 (a) Flute

8

10

n

0

2

4

6

8

10

n

(b) Violin

In addition to analyzing the sounds of conventional musical instruments, Fourier series enable us to synthesize sounds. The idea behind music synthesizers is that we can combine various pure tones (harmonics) to create a richer sound through emphasizing certain harmonics by assigning larger Fourier coefﬁcients (and therefore higher corresponding energies).

||||

1–6

||||

Exercises

7–11

||||

, A function f is given on the interval and f is periodic with period 2 . (a) Find the Fourier coefﬁcients of f . (b) Find the Fourier series of f . For what values of x is f x equal to its Fourier series? ; (c) Graph f and the partial sums S2, S4, and S6 of the Fourier series.

1. f x

Find the Fourier series of the function. 1 0 0 1 0 0 1 if x if 1 1 x 2 f x 4 f x

7. f x

8. f x

1 1 0 x x x2

if if 0 if if 0 x

x x x 0

0

9. f x 10. f x 11. f t s s

if 2 x 0 if 0 x 1 if 1 x 2 x if 4 x 0 if 0 x 4 x, 1 1 s f x

4

f x

f x f x

8 2

f x f x

2. f x 3. f x 4. f x 5. f x

x t s 1 1 s sin 3 t , s s

s

s

s

s

s

0 if cos x if 0 1 1 0 s s

x x x 2 x x s 0 2 0 s s s s s s

12. A voltage E sin

t, where t represents time, is passed through a so-called half-wave rectiﬁer that clips the negative part of the wave. Find the Fourier series of the resulting periodic function 0 if if 0 t t 0 f t E sin t 2 f t

6. f x s s

if if if 0 s f t

10

❙❙❙❙

FOURIER SERIES

13–16 |||| Sketch the graph of the sum of the Fourier series of f without actually calculating the Fourier series.

18. Use the result of Example 2 to show that

13. f x

1 3 x 1 x , e x, s s

if 4 x 0 if 0 x 4 if 1 x 0 x if 0 x 1 1 2 x x s 1

1 32

1 52

1 72

2

8

19. Use the result of Example 1 to show that

14. f x 15. f x 16. f x s s

1

1 3

1 5

1 7

4

3

1 2 s s s s s s s

20. Use the given graph of f and Simpson’s Rule with n

8 to estimate the Fourier coefﬁcients a 0, a1, a 2, b1, and b2. Then use them to graph the second partial sum of the Fourier series and compare with the graph of f . y 17. (a) Show that, if

1 1 3

x

1, then 1 n x2

n 1

4 n2

2

cos n x

1

(b) By substituting a speciﬁc value of x, show that 1 n2

2

0.25

x

n 1

6…...

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...1 | (Radians/Real Numbers) | Input | 26·3π | Output | 1π | Decimal Output | 0.3183098861838 | 2 | (Radians/Real Numbers) | Input | 26·3π | Output | 1π | Decimal Output | 0.3183098861838 | 3 | (Radians/Real Numbers) | Input | 1π 4 sinπ3·2+6π cosπ3·2-6π | Output | 2 3-9π | Decimal Output | -1.7621311848105 | 4 | (Radians/Real Numbers) | Input | 1π 4sinπ3·2+6πcosπ3·2-1π 6π | Output | 2 3π-9π2 | Decimal Output | 0.1907671380625 | 5 | (Radians/Real Numbers) | Input | 1π 6-2sinπ3·2-3πcosπ3·2-1π 6-6sinπ3·6-3πcosπ3·6 | Output | 92 π2+2 3π | Decimal Output | 1.5586031172341 | 6 | (Radians/Real Numbers) | Input | 1π 4sinπ3·2+6πcosπ3·2-1π 6π+1π 6-2sinπ3·2-3πcosπ3·2-1π 6-6sinπ3·6-3πcosπ3·6 | Output | 4 3π-92 π2 | Decimal Output | 1.7493702552966 | 7 | (Radians/Real Numbers) | Input | 26 6-x sinπ3 x π3 -sinπ3 x π3 -1ⅆx | Output | 6-xsinπ x3π+9cosπ x3π2+x-С | 8 | (Radians/Real Numbers) | Input | 26 6-2 sinπ3·2 π3 -sinπ3·2 π3 -1ⅆx | Output | -3 3 x2+2 3π+x-С | 9 | (Radians/Real Numbers) | Input | 1π 6-xsinπ3 x-3πcosπ3 x limx→2-6-1π 6-xsinπ3 x-3πcosπ3 x | Output | -32 π+5 3 6-xsinπ x3-3cosπ x3ππ2 | 10 | (Radians/Real Numbers) | Input | 1π 6-2sinπ3·2-3πcosπ3·2-1π 6-6sinπ3·6-3πcosπ3·6 | Output | 92 π2+2 3π | Decimal Output | 1.5586031172341 | 11 | (Radians/Real Numbers) | Input | 1π 4sinπ3·2+6πcosπ3·2-6π+1π 6-2sinπ3·2-3πcosπ3·2-1π 6-6sinπ3·6-3πcosπ3·6 | Output | 4 π 3+92-9 ππ2 | Decimal......

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...Fourier Transform and its applications Jatin Kumar Murray State University Abstract It has been widely recognized that waveforms are an integral part of the various universe phenomenon. Waveforms can be used to represent almost everything in the world. Therefore it is understandable that concepts related to waveforms or signals are extremely important as their applications exist in a broad variety of fields. The processes and ideas related to waveforms play a vital role in different areas of science and technology such as communications, optics, quantum mechanics, aeronautics, image processing to name a few. Even though the physical nature of signals might be completely different in various disciplines, all waveforms follow one fundamental principle; they can be represented by functions of one or more independent variables. This paper would focus on the concept of Fourier Transform, the technique through which signals can be deconstructed and represented as sum of various elementary signals. It briefly describes Linear Time Invariant systems and their response to superimposed signals. Fourier transform has many applications in physics and Engineering. This paper would also cover some of Fourier Transform applications in telecommunication and its impact on society. Introduction Some of the basic signals that exist in the world and are useful in various technology fields are continuous and discrete......

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...policy in this regard. 2. I declare that this report is my own original work. Where other people’s work has been used (either from a printed source, Internet or any other source), this has been properly acknowledged and referenced in accordance with departmental requirements. 3. I have not used work previously produced by another student or any other person to hand in as my own. 4. I have not allowed, and will not allow, anyone to copy my work with the intention of passing it off as his or her own work. Name | Student number | Signature | | | | Date | | Contents Table of Figures 3 Acronyms Used 3 PART 1 Filtering in the Frequency Domain 3 Introduction 3 Review of Prior Knowledge 4 Complex Numbers 4 Fourier series 4 Fourier Transform 4 Convolution Theorem 5 Overview 5 2-D FFT 6 DFT 6 IDFT 7 2-D FFT 7 Comparison with 1-D FFT 8 2-D FFT and Image Processing 8 Image Smoothing and Sharpening 9 Smoothing 9 Sharpening 11 Conclusion 13 PART 2 Application of Filtering in the Frequency Domain 13 Introduction 13 Gaussian Filter Theoretical Analysis 13 Gaussian Low Pass Filter 14 Gaussian High Pass Filter 14 Gaussian Filter Design 15 Practical Results 16 Conclusion 17 References 18 Table of Figures Figure 1: Input Signal Transformed to Frequency Domain 5 Figure 2: Flow Diagram of Filtering in the Frequency Domain 5 Figure 3: Spatial VS Frequency Domain 5 Figure 4: 2-D Sinusoidal Wave 7 Figure 5: Flow......

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...in textbooks on Fourier theory. It seems that there is little that can be done with wavelets that cannot be done with traditional Fourier analysis. Stephane Mallat was not the father of wavelet theory, but he is certainly an evangelist. His textbook on the subject, A Wavelet Tour of Signal Processing [1], contains proofs about the theory of wavelets, and a summation about what is known about them with applications to signal processing. One of his many papers, Characterization of Signals from Multiscale Edges [2], is frequently cited as a link between wavelets and edge detection. Mallat’s method not only finds edges, but classifies them into different types as well. Mallat goes on to describe a method of recovering complete images using only the edges, but we will not implement it in this project. In this project, we study this paper, and implement the method of Mallat to multiscale edge detection and analysis. We will first present a short background on wavelet theory. Then we will describe the different types of edges that exist in images, and how they can be characterized using a Lipschitz constant. Next, we describe the algorithm for the wavelet transform, from the Mallat paper. Finally, we show the results of applying the algorithm to a test image, and a real image. wave with the signal. When the results high valued, the coefficients of the Fourier transform will be high. Where the signal or the wave is close to zero, the coefficients will be low. Fourier analysis has......

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...complex part is multiplied by the imaginary number j, where j is the square root of -1. (In other branches of applied mathematics i is usually used to represent the imaginary number, however in electrical engineering j is used because the letter i is used to denote electrical current.) For the complex number: a + jb (58) (57) 64 DSPedia a is the real part, where a ∈ ℜ (ℜ is the set of real numbers) and jb is the imaginary part, where b ∈ ℜ . Complex arithmetic can be performed and the result expressed as a real part and imaginary part. For addition: ( a + jb ) + ( c + jd ) = ( a + c ) + j ( b + d ) and for multiplication: ( a + jb ) ( c + jd ) = ( ac – bd ) + j ( ad + bc ) (60) (59) Complex number notation is used to simplify Fourier analysis by allowing the expression of complex sinusoids using the complex exponential e jω = cos ω + j sin ω . Also in DSP complex numbers represent a convenient way of representing a two dimensional space, for example in an adaptive beamformer (two dimensional space), or an adaptive decision feedback analyser where the inphase component is a real number, and the quadrature phase component is a complex number. See also Complex Conjugate, Complex Sinusoid. Complex Plane: The complex plane allows the representation of complex numbers by plotting the real part of a complex number on the x-axis, and the imaginary part of the number on the y-axis. Imaginary, ℑ 4 3 2 1 -4 -3 -2 -1 -1 0 1 2 3 4 2 + 3j 3 Real, ℜ -3.51-......

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...ENTS 699R: Lecture 1d support ENTS 699R Lecture 1d support: Fourier Transform Tables Alejandra Mercado June, 2013 1 Transform Pairs The following is a table of basic transform pairs that can be used as building blocks to derive more complicated transform pairs: Time domain function, with dummy variable t 1 2 3 4 5 6 7 F F F Frequency domain function, with dummy variable f δ(t) ⇐⇒ 1 1 ⇐⇒ δ(f ) δ(t − t0 ) ⇐⇒ e−j2πf t0 sin (2πf0 t + φ) ⇐⇒ F F j −jφ δ(f 2 [e 1 −jφ δ(f 2 [e + f0 ) − ejφ δ(f − f0 )] cos (2πf0 t + φ) ⇐⇒ + f0 ) + ejφ δ(f − f0 )] 1 |t| ≤ T F t 2 rect ( T ) = ⇐⇒ T sinc(f T ) = T sin(πf T ) πf T 0 o.w. sinc(βt) ⇐⇒ F 1 β f · rect ( β ) = 1 β ·1 |f | ≤ o.w. β 2 0 Page 1 ENTS 699R: Lecture 1d support 2 Properties For the table of Fourier Transform properties, assume that we already know that: g(t) ⇐⇒ G(f ) h(t) ⇐⇒ H(f ) and that α, β, T, φ, f0 , t0 are all arbitrary constants. Time domain function, with dummy variable t A B C D E F F G H I F F F F F Frequency domain function, with dummy variable f Property name time/frequency reversal duality time shift frequency shift linearity g(−t) ⇐⇒ G(−f ) G(t) ⇐⇒ g(−f ) g(t − t0 ) ⇐⇒ g(t)ej2πf0 t F F F e−j2πf t0 G(f ) ⇐⇒ G(f − f0 ) 1 2 j 2 αg(t) + βh(t) ⇐⇒ αG(f ) + βH(f ) g(t) cos(2πf0 t) ⇐⇒ g(t) sin(2πf0 t) ⇐⇒ F F F (G(f − f0 ) + G(f + f0 )) modulation (G(f + f0 ) − G(f − f0 )) modulation multipl. in time domain convolution in time domain time scaling g(t) ×......

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...Question 3.2) i) Fourier series A Fourier series is a type of infinite series in mathematicswhich involves functions of trigonometry. Fourier series are used in applied mathematics, physics and electronics, in order to express periodic functions comprising communication signal waveforms.Many of the phenomena studied in engineering and science are periodic in nature for example, the current and voltage in an alternating current circuit. These periodic functions can be analyzed into their constituent components (fundamentals and harmonics) by a process called Fourier analysis (Asher, 2013). The first thing that comes under notice is representation of Fourier series for a function that is periodic in nature. A function f(x) is said to be Periodic when f(x + T) = f(x) for some value T. The minimum value of T is called the Fundamental Period. A notable feature of the periodic function is that if two functions are added or multiplied that has the same period, then the resultant sum or product will be of the same period (Combined estimator Fourier series and spline truncated in multivariable nonparametric regression, 2015). For example, cos(2πt), sin(2πt) both have periods of 1. The sums of these two functions are: cos(2π(t + 1)) + sin(2π(t + 1)) = cos(2πt + 2π) + sin(2πt + 2π) = cos(2πt) + sin(2πt) It is observed that the resultant is of the same period as that of the initial trigonometric functions is of period 1. The representation of the Fourier series of a......

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