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Chapter 7: Q5P (page 347)

Using the definition of a periodic function, show that a sum of terms corresponding to a fundamental musical tone and its overtones has the period of the fundamental.

Short Answer

Expert verified

The sum of sine and cosine terms is f(t).

Step by step solution

01

Given information

Show that the sum of terms corresponding to a musical tone and its overtone has the period of fundamental.

02

Step 2: Definition of a periodic function

Periodic function can be defined as a function returning to the same value at regular intervals.

03

Step 3: Sum of terms corresponding to a fundamental musical tone and its overtones

Let the function f(t) be a sum of a sine fundamental term and its overtones as well as the cosine terms:

f(t)=鈭�k=1nAksin(蝇kt)+Bkcos(蝇kt)

If,.f(t)=f(T+t)

Then, obtainf(t+T)=鈭�k=1nAksin(蝇kt+2蟺k)+Bkcos(蝇kt+2蟺k)

Where,.T = 2蟺k

=鈭�k=1nAksin(蝇kt)+Bkcos(蝇kt)=f(t)

Thus, f(t) is the sum of sine and cosine terms.

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Most popular questions from this chapter

Show that if (12.2) is written with the factor 12蟺multiplying each integral, then the corresponding form of Parseval鈥檚 (12.24) theorem is 鈭�-鈭�鈭�|f(x)|2dx=鈭�-鈭�鈭�|g(伪)|2d伪.

For each of the periodic functions in Problems 5.1 to 5.11 , use Dirichlet's theorem to find the value to which the Fourier series converges atx=0,卤蟺/2,卤蟺,卤2蟺 .

In each of the following problems you are given a function on the interval -蟺<x<蟺. Sketch several periods of the corresponding periodic function of period2蟿蟿 . Expand the periodic function in a sine-cosine Fourier series,

role="math" localid="1659239194875" f(x)={1,-蟺<x<蟺2and0<x<蟺20,-蟺2<x<0and蟺2<x<蟺

Represent each of the following functions (a) by a Fourier cosine integral; (b) by a Fourier sine integral. Hint: See the discussion just before Parseval鈥檚 theorem.

28.f(x)={1,2<x<40,0<x<2,x>4

Sketch several periods of the corresponding periodic function of period 2蟿蟿 . Expand the periodic function in a sine-cosine Fourier series.

role="math" localid="1659236419546" f(x)={0,-蟺<x<0-1,0<x<蟺21,蟺2<x<蟺
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