Unit 6: Introduction to Fourier series - Subjective Questions

A function of period can be represented by the Fourier series

Using the orthogonality of sine and cosine functions on :

• Integrating the series over gives
  
• Multiplying by and integrating gives
  
• Multiplying by and integrating gives
  

These coefficient relations are called Euler's formulae.

For positive integers and , the required orthogonality relations are:

and

Also,

These relations ensure that multiplication and integration isolate one Fourier coefficient at a time.

Euler's exponential identities are

Therefore,

Using these identities, a Fourier series of period can be written as

where

The real and complex coefficients are related by

The complex form is compact and is especially useful in engineering applications involving frequency-domain analysis.

A periodic function has a convergent Fourier expansion under the following sufficient Dirichlet conditions over any one period:

• must be absolutely integrable:
  
• It must have only a finite number of maxima and minima.
• It must have only a finite number of finite discontinuities.
• It must be single-valued and finite, except possibly at a finite number of points.

If these conditions hold, the Fourier series converges:

• To at every point where is continuous.
• To the mean of the one-sided limits at a jump discontinuity:
  

The conditions are sufficient but not necessary; some functions that do not satisfy every condition may still possess Fourier expansions.

Suppose a piecewise smooth periodic function has a jump discontinuity at . Let

At , its Fourier series converges to

Important observations are:

• The actual assigned value does not affect the Fourier coefficients because changing a function at a finite number of points does not alter its integrals.
• If is continuous at , then , so the series converges to .
• At the endpoints of a chosen interval, the one-sided limits are determined from the periodic extension of the function.

The function is odd, so

The sine coefficients are

Thus,

Hence the Fourier series is

At and at the periodically identified endpoints , the left- and right-hand limits are and . Therefore, the series converges there to

The Gibbs phenomenon is the oscillatory overshoot and undershoot that occurs when a Fourier series is truncated near a jump discontinuity.

• As more terms are included, the oscillations become confined to a narrower region around the discontinuity.
• The maximum overshoot does not approach zero; it approaches approximately of the jump size.
• Away from the discontinuity, the partial sums approach the function more accurately.
• At the discontinuity itself, the Fourier series converges to
  

Thus, adding terms improves localization but does not eliminate the limiting relative overshoot. This effect is important in signal processing when discontinuous signals are reconstructed from a finite number of harmonics.

Let the interval length be

Introduce the change of variable

Then corresponds to , and corresponds to . The Fourier expansion becomes

The coefficients are

This procedure is called a change of interval. The resulting series represents the periodic extension of with period .

Use the substitution

When , , and when , . Define

The standard Fourier series of is

Replacing by gives

The corresponding coefficients are

For a period of , take . The Fourier series has the form

The coefficients are

and, by integration by parts,

Therefore,

At and , the periodic extension has limiting values and , so the series converges to their mean:

A function is even if

and it is odd if

For an even function on :

• is odd.
• Therefore, .
• The remaining coefficients are
  
  

For an odd function on :

• and .
• The sine coefficients are
  

Thus, even functions have cosine series, while odd functions have sine series.

Even-function Fourier series:

• Condition: .
• Contains a constant term and cosine terms only.
• Its form is
  
• All sine coefficients vanish: .

Odd-function Fourier series:

• Condition: .
• Contains sine terms only.
• Its form is
  
• The coefficients and vanish.

The simplification follows from the facts that the integral of an odd function over is zero and the integral of an even function is twice its integral over .

Since is odd,

The sine coefficients are

Integrating by parts,

Therefore,

Hence,

Equivalently,

At , the periodic extension has a jump from to , so the series converges to .

The function is even, so . The constant coefficient is

For ,

Integrating twice by parts gives

Thus,

Therefore,

The periodic extension is continuous at because both endpoint values are .

Since is even, . The constant coefficient is

For ,

Therefore,

Hence,

Only odd cosine harmonics occur because the even-indexed coefficients vanish.

Suppose is given only on .

A half-range sine series is obtained by extending as an odd function to :

Its expansion is

where

A half-range cosine series is obtained by extending as an even function:

Its expansion is

where

The half-range sine series is

where

Let . Integration by parts gives

Thus,

Therefore,

for . This series corresponds to the odd periodic extension of .

The cosine coefficients are

and

After integration,

Hence for even , while

for odd . Therefore,

This represents the even periodic extension of .

The sine coefficients are

Evaluating the integral gives

Therefore,

The half-range sine series is

for . At and , the odd periodic extension is discontinuous and the series has value .

The half-range cosine series is

The constant coefficient is

For ,

Integrating twice by parts gives

Therefore,

This series is generated by the even extension on .