Section 4 The Fourier Series and Fourier Transform

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Section 4 The Fourier Series and Fourier Transform

Fourier Series Representation of Periodic Signals Let x ( t ) be a CT occasional flag with period T, i.e., Example: the rectangular heartbeat prepare

The Fourier Series Then, x ( t ) can be communicated as where is the key recurrence ( rad/sec ) of the flag and is known as the consistent or dc segment of x ( t )

Dirichlet Conditions An intermittent flag x ( t ), has a Fourier arrangement on the off chance that it fulfills the accompanying conditions: x ( t ) is totally integrable over any period, to be specific x ( t ) has just a limited number of maxima and minima over any period x ( t ) has just a limited number of discontinuities over any period

Example: The Rectangular Pulse Train From figure , so Clearly x ( t ) fulfills the Dirichlet conditions and in this way has a Fourier arrangement representation

Example: The Rectangular Pulse Train – Cont'd

Trigonometric Fourier Series By utilizing Euler's recipe, we can revamp as the length of x(t) is genuine This expression is known as the trigonometric Fourier arrangement of x ( t ) dc segment k - th consonant

Example: Trigonometric Fourier Series of the Rectangular Pulse Train The expression can be modified as

Gibbs Phenomenon Given an odd positive whole number N, characterize the N - th halfway entirety of the past arrangement According to Fourier's hypothesis , it ought to be

Gibbs Phenomenon – Cont'd

Gibbs Phenomenon – Cont'd overshoot : around 9 % of the flag greatness (exhibit regardless of the possibility that )

Parseval's Theorem Let x ( t ) be an occasional flag with period T The normal influence P of the flag is characterized as Expressing the flag as it is likewise

Fourier Transform We have seen that intermittent signs can be spoken to with the Fourier arrangement Can aperiodic signs be broke down as far as recurrence parts? Yes, and the Fourier change gives the apparatus to this investigation The real distinction w.r.t. the line spectra of occasional signs is that the spectra of aperiodic signs are characterized for every genuine estimation of the recurrence variable not only for a discrete arrangement of qualities

Frequency Content of the Rectangular Pulse

Frequency Content of the Rectangular Pulse – Cont'd Since is intermittent with period T, we can compose where

Frequency Content of the Rectangular Pulse – Cont'd What happens to the recurrence parts of as ? For

Frequency Content of the Rectangular Pulse – Cont'd plots of versus for

Frequency Content of the Rectangular Pulse – Cont'd It can be effectively demonstrated that where

Fourier Transform of the Rectangular Pulse The Fourier change of the rectangular heartbeat x ( t ) is characterized to be the point of confinement of as , i.e.,

The Fourier Transform in the General Case Given a flag x ( t ), its Fourier change is characterized as A flag x ( t ) is said to have a Fourier change in the customary sense if the above indispensable joins

The Fourier Transform in the General Case – Cont'd The essential converges if the flag x ( t ) is " very much carried on " and x ( t ) is totally integrable , in particular, Note: all around acted implies that the flag has a limited number of discontinuities, maxima, and minima inside any limited time interim

Example: The DC or Constant Signal Consider the flag Clearly x ( t ) does not fulfill the main necessity since Therefore, the steady flag does not have a Fourier change in the standard sense Later on, we'll see that it has however a Fourier change in a summed up sense

Example: The Exponential Signal Consider the flag Its Fourier change is given by

Example: The Exponential Signal – Cont'd If , does not exist If , and does not exist either in the normal sense If , it is abundancy range stage range

Example: Amplitude and Phase Spectra of the Exponential Signal

Rectangular Form of the Fourier Transform Consider Since as a rule is a perplexing capacity, by utilizing Euler's recipe

Polar Form of the Fourier Transform can be communicated in a polar shape as where

Fourier Transform of Real-Valued Signals If x ( t ) is genuine esteemed, it is Moreover whence Hermitian symmetry

Example: Fourier Transform of the Rectangular Pulse Consider the even flag It is

Example: Fourier Transform of the Rectangular Pulse – Cont'd

Example: Fourier Transform of the Rectangular Pulse – Cont'd plentifulness range stage range

Bandlimited Signals A flag x ( t ) is said to be bandlimited if its Fourier change is zero for all where B is some positive number, called the data transmission of the flag It would seem any bandlimited flag must have an unbounded length in time, i.e., bandlimited signals can't be time restricted

Bandlimited Signals – Cont'd If a flag x ( t ) is not bandlimited, it is said to have boundless transfer speed or an unending range Time-constrained signs can't be bandlimited and in this way unequaled constrained signs have vast transmission capacity However, for any all around acted flag x ( t ) it can be demonstrated that whence it can be accepted that B being a helpful substantial number

Inverse Fourier Transform Given a flag x ( t ) with Fourier change , x ( t ) can be recomputed from by applying the backwards Fourier change given by Transform match

Properties of the Fourier Transform Linearity: Left or Right Shift in Time: Time Scaling:

Properties of the Fourier Transform Time Reversal: Multiplication by a Power of t: Multiplication by a Complex Exponential:

Properties of the Fourier Transform Multiplication by a Sinusoid (Modulation): Differentiation in the Time Domain:

Properties of the Fourier Transform Integration in the Time Domain: Convolution in the Time Domain: Multiplication in the Time Domain:

Properties of the Fourier Transform Parseval's Theorem: Duality: if

Properties of the Fourier Transform - Summary

Example: Linearity

Example: Time Shift

Example: Time Scaling time pressure recurrence development time extension recurrence pressure

Example: Multiplication in Time

Example: Multiplication in Time – Cont'd

Example: Multiplication by a Sinusoid sinusoidal burst

Example: Multiplication by a Sinusoid – Cont'd

Example: Integration in the Time Domain

Example: Integration in the Time Domain – Cont'd The Fourier change of x ( t ) can be effortlessly observed to be Now, by utilizing the reconciliation property, it is

Example: Integration in the Time Domain – Cont'd

Generalized Fourier Transform Fourier change of Applying the duality property summed up Fourier change of the consistent flag

Generalized Fourier Transform of Sinusoidal Signals

Fourier Transform of Periodic Signals Let x ( t ) be an occasional flag with period T ; thusly, it can be spoken to with its Fourier change Since , it is

Fourier Transform of the Unit-Step Function Since utilizing the combination property, it is

Common Fourier Transform Pairs