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Chapter 3, Problem 22P

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Signals and Systems (2nd Edition)

Problem

Determine the Fourier series representations for the following signals: (a) Each x(t) illustrated in Figure P3.22(a)--(f). (b) x(t) periodic with period 2 and

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TEXTBOOK SOLUTIONS 

Chapter 3, Problem 22P

Chapter 1 Chapter 2 Chapter 3 1P 2P 3P 4P 5P 6P 7P 8P 9P 10P 11P 12P 13P 14P 15P 16P 17P 18P

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19P

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20P 21P

Chapter 3, Problem 22P

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22P 23P 24P 25P 26P 27P 28P 29P 30P 31P 32P 33P 34P

Step-by-step solution

35P

Step 1 of 24

36P 37P

(a) (a) Consider the following waveform: 38P 39P 40AP 41AP 42AP 43AP 44AP 45AP

Figure 1

46AP 47AP

Comment

48AP

Step 2 of 24

49AP 50AP 51AP

Determine the Fourier series representation for the signal in Figure 1. Clearly, from the waveform in Figure 1:

52AP 53AP

Calculate the dc component of

54AP

The fundamental period,

55AP

Calculate

56AP

.

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57AP

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58AP 59AP

Chapter 3, Problem 22P

60AP 61AP 62AP

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Since, the waveform in Figure 1 is symmetrical about t-axis (X-axis), therefore, the dc component of

is:

Determine the Fourier series coefficients of

63AP 64AP 65EP 66EP 67EP 68EP 69EP

Comment 70EP 71EP

Step 3 of 24

Chapter 4 Chapter 5

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Simplify further.

Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10 Chapter 11

Hence, the Fourier series representation of

Thus, the Fourier series representation of

is:

is

.

Comment

Step 4 of 24 (a) (b) Consider the following waveform:

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TEXTBOOK SOLUTIONS 

Chapter 3, Problem 22P

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Figure 2

Comment

Step 5 of 24 Determine the Fourier series representation for the signal in Figure 2. Clearly, from the waveform in Figure 2: The fundamental period, Calculate

For interval

,

is defined as:

Calculate the dc component of

Simplify further.

Comment

.

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Step 6 of 24 EXPERT Q&A

Determine the Fourier series coefficients of

Chapter 3, Problem 22P

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Simplify further.

Comment

Step 7 of 24 Simplify further.

Simplify further.

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Chapter 3, Problem 22P

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Comment

Step 8 of 24 Since,

Hence,

Thus, the Fourier series representation of

is

Comment

Step 9 of 24 (a) (c) Consider the following waveform:

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Chapter 3, Problem 22P

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Figure 3

Comment

Step 10 of 24

Determine the Fourier series representation for the signal in Figure 3. Clearly, from the waveform in Figure 3: The fundamental period, For interval

,

is defined as:

Calculate the dc component of

.

Simplify further.

Hence, the dc component of

is:

Comment

Step 11 of 24

Determine the Fourier series coefficients of

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Simplify further.

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TEXTBOOK SOLUTIONS 

Chapter 3, Problem 22P

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Comment

Step 12 of 24 Simplify further.

Thus, for

Comment

Step 13 of 24 (a) (d) Consider the following waveform:

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Chapter 3, Problem 22P

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Figure 4

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Step 14 of 24 Determine the Fourier series representation for the signal in Figure 4. Clearly, from the waveform in Figure 4: The fundamental period, For interval

,

is defined as:

Calculate the dc component of

Hence, the dc component of

.

is:

Determine the Fourier series coefficients of

Where,

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Chapter 3, Problem 22P

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Hence,

Thus, for

Comment

Step 15 of 24 (a) (e) Consider the following waveform:

Figure 5

Comment

Step 16 of 24 Determine the Fourier series representation for the signal in Figure 5. Clearly, from the waveform in Figure 5: The fundamental period, For interval

,

is defined as:

Calculate the dc component of

.

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Chapter 3, Problem 22P

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Hence, the dc component of

is:

Comment

Step 17 of 24

Determine the Fourier series coefficients of

Simplify further.

Thus,

.

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TEXTBOOK SOLUTIONS  Comment

Chapter 3, Problem 22P

Step 18 of 24

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(a) (f) Consider the following waveform:

Figure 6

Comment

Step 19 of 24

Determine the Fourier series representation for the signal in Figure 6. Clearly, from the waveform in Figure 6: The fundamental period, For interval

,

The fundamental period of

Hence, the dc component of

is defined as:

is

is:

Comment

Step 20 of 24

Determine the Fourier series coefficients of

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Chapter 3, Problem 22P

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Simplify further.

Simplify further.

Thus, .

Comment

Step 21 of 24

(b) Consider the following signal: for The period of

is:

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Determine the Fourier series coefficients of TEXTBOOK SOLUTIONS EXPERT Q&A

 Chapter 3, Problem 22P

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Simplify further.

Thus, .

Comment

Step 22 of 24 (c) Consider the following signal:

The period of

is:

Calculate the dc component of the signal.

Simplify further.

Hence, the dc component of

is:

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TEXTBOOK SOLUTIONS  Comment

Chapter 3, Problem 22P

Step 23 of 24

Determine the Fourier series coefficients of

Simplify further.

Comment

Step 24 of 24 Simplify further.

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Chapter 3, Problem 22P

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Simplify further.

Thus, .

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Chapter 3, Problem 22P

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