Give the Fourier coefficients a0, an, and bn of the waveform in Fig. 16.47 . Plot the amplitude

step 1 Hello everyone, let us do the following question. Here we have given the Fourier coefficient, A

Give the Fourier coefficients a0, an, and bn of the waveform...

Key Concepts

Orthogonality of Trigonometric Functions

Orthogonality is a fundamental property of sine and cosine functions over a specified interval, typically one period of the signal. This property ensures that each term in the Fourier series is independent of the others, allowing for the unique calculation of each Fourier coefficient via integration. It is essential for decomposing a complex signal into its harmonic components without overlap.

Amplitude Spectrum

The amplitude spectrum is a graphical representation that shows the magnitude of each Fourier coefficient as a function of frequency. It provides insight into how much each harmonic contributes to the overall signal and helps in identifying the dominant frequencies present in the waveform.

Phase Spectrum

The phase spectrum displays the phase shift associated with each harmonic component in the Fourier series. By representing the angle (or phase) of each Fourier coefficient, typically expressed in radians, it provides additional information about the timing and alignment of the sine and cosine components relative to the original signal.

Fourier Series

The Fourier series is a method used to represent a periodic function as an infinite sum of sines and cosines. This expansion enables the analysis of complex periodic waveforms by breaking them down into their constituent harmonic components, which simplifies understanding of the signal’s frequency content.

Fourier Coefficients

Fourier coefficients, typically denoted as a?, a?, and b? for the sine and cosine terms, are the weights assigned to the corresponding trigonometric functions in a Fourier series. These coefficients are determined through integration over one period of the signal and capture the amplitude of each harmonic component, thereby forming the basis for reconstructing the original waveform.

Transcript

00:01 Hello everyone, let us do the following question.

00:02 Here we have given the fourier coefficient, a node, a1 and bn of the waveform.

00:07 We have to plot the amplitude and face spectrum.

00:11 We have to plot the amplitude and face spectrum.

00:15 Here we will say that time is given as 4 and omega node is given as 2 pi by t.

00:23 T is given as 4.

00:24 So this will be pi by 2.

00:27 It will write value for g at any time t.

00:30 This will be curves to 5 4 t if we'll write this we will write 4 0 to t to 1 when t lies between 0 to 1 and this will be 10 when this lies between 1 and 2 when this lies between 1 and 2 this will be 0 when this lies between 2 and 4 when this lies between 2 and 4 now if we'll find for a nod here we will have 1 by t here here we will write 0 to t here we will write 0 to t into g at any time t into d t here we will write 2 .5 here we will write this 0 to 1 into 5 into d t plus 1 to 2 10 into d t after we will solve we will get this equation as 3 .75.

01:28 3 .75.

01:30 Now, if we'll write this 4 an, if we will write this 4 an, here we will get this as 2 by t.

01:38 From this we will get this equation as 0 to t.

01:43 This is g at any time t.

01:46 This is cos and omega node t into dt.

01:52 Cos and mega -knot t into dt.

01:54 From this we will get 2 by 4.

01:58 This is 0 to 1.

02:00 Here we will write this equation as 5 cos, n pi by 2t into d t.

02:08 Plus here we will get this as 1 to 2.

02:11 10, cos, n pi by 2 t into d t.

02:17 After we will solve this equation as 0 .5.

02:22 5 5 5 2 by n pi into sine n pi by 2 t from 0 to 1 here we will get 10 into 2 divided by n pi this is sign n pi by 2 into t this will be 1 to 2 after we will follow this equation, we will get this as minus 1 by n pi into 5, sine n pi by 2.

03:03 So we will write value for a .n, this is 5 by n pi.

03:09 Here we will write minus 1, n plus 1 by 2 when n plus 1 by 2 when n is owed, and this will be 0.

03:20 When n is even.

03:23 This will be 0 when n is even.

03:26 And it will write value for vn.

03:31 This is 2 by t.

03:33 This is 0 to t.

03:36 Gt into sign and n omega node t into dt.

03:43 This is 2 by 4 from 0 to t.

03:48 This is 5.

03:51 N pi by 2 into t into d t from this it will solve this is 0 to 1 further it will solve this from 1 to 2 if we'll solve this from 1 to 2 this is written as 10 sign n pi by 2 t into d t after we will solve this equation we will get this 5 by n pi by n pi 2 x0c0 n pi plus cose n pi by so now if we'll put further values if we will put further values of n a n b n and phase we will get here we will write the terms here we will write n here we will write n here we will write bn here we will write capital a n here we will write face...

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