Fourier series: review

Fourier series: review

A function or signal $x\left(t\right)$ is called periodic with period $T$ if $x(t+T)=x\left(t\right)$ . All “typical” periodic function $x\left(t\right)$ with period $T$ can be developed as follows

where the coefficients are computed as follows:

The natural interpretation of [link] is as a decomposition of the signal $x\left(t\right)$ into individual oscillations where $a_k$ indicates the amplitude of the even oscillation $\text{cos}\left(k\frac{2\pi }{T}t\right)$ of frequency $k/T$ (meaning its period is $T/k$ ), and $b_k$ indicates the amplitude of the odd oscillation $\text{sin}\left(k\frac{2\pi }{T}t\right)$ of frequency $k/T$ . For an audio signal $x\left(t\right)$ , frequency corresponds to how high a sound is and amplitude to how loud it is. The oscillations appearing in the Fourier decomposition are oftenalso called harmonics (first, second, third harmonic etc).

Note: one can also integrate over $[0,T]$ or any other interval of length $T$ . Note also, that the average value of $x\left(t\right)$ over one period is equal to $a_0/2$ .

Complex representation of Fourier series

Often it is more practical to work with complex numbers in the area of Fourier analysis. Using the famous formula

it is possible to simplify several formulas at the price of working with complex numbers.Towards this end we write

From this we observe that we may replace the cos and sin harmonics by a pair of exponential harmonicswith opposite frequencies and with complex amplitudes which are conjugate complex to each other.In fact, we arrive at the more simple complex Fourier series :

Note that $X$ is complex, but $x$ is real-valued (the imaginary parts of all the terms in $\sum X_k{e}^{j2\pi kt/T}$ add up to zero; in other words, they cancel each other out). The absolute value of $X_k$ gives the amplitude of the complex harmonic with frequency $k/T$ (meaning its period is $T/k$ ); the argument of $X_k$ provides the phase difference between the complex harmonics.If $x$ is even, $X_k$ is real for all $k$ and all harmonics are in phase.

To verify [link] note that by [link] and [link] we have for positive $k$

For negative $k$ we note that $X_{-k}=X_k^{*}$ by [link] , where ${\left(\right)}^{*}$ denotes the conjugate complex. By [link] , the $X_k$ are exactly as they are supposed to be.

Properties

• Linearity: The Fourier coefficients of the signal $z\left(t\right)=cx\left(t\right)+y\left(t\right)$ are simply

• Change of frequency: The signal $z\left(t\right)=x\left(\lambda t\right)$ has the period $T/\lambda$ and has the same Fourier coefficients as $x\left(t\right)$ — but they correspond to different frequencies $f$ :

The equation on the right allows to read off the Fourier coefficients and to establish $Z_k=X_k$ . (For an alternative computation see  Comment 1 )

Comment 1

• Shift: The Fourier coefficients of $z\left(t\right)=x(t+d)$ are simply

The modulation is much more simple in complex writing then it would be with real coefficients.For the special shift by half a period, i.e., $d=T/2$ we have $Z_k=X_k{e}^{j\pi k}={(-1)}^k{X}_k$ .

• Derivative: The Fourier series of the derivative of $x\left(t\right)$ with development [link] can be obtained simply by taking the derivative of [link] term by term:

Short: when taking the derivative of a signal, the complex Fourier coefficients get multiplied by $k\frac{2\pi j}{T}$ . Consequently, the coefficients of the derivative decay slower.

Examples

• The pure oscillation (containing only one real but two complex frequencies)
  $$x\left(t\right)=\text{sin}(2\pi t/T)X_1=-X_{-1}=\frac{j}{2}$$
  or $B_1=1/2=-B_{-1}$ , or $b_1=1$ , and all other coefficients are zero.This formula can be obtained without computing integrals by noting that $\text{sin}\left(\alpha \right)=(e^{j\alpha }-e^{-j\alpha })/\left(2j\right)=(j/2)(e^{-j\alpha }-e^{j\alpha })$ and setting $\alpha =2\pi t/T$ .
• Functions which are time-limited , i.e., defined on a finite interval can be periodically extended. Example with $T=2\pi$ :
  $$x\left(t\right)=\left\{\begin{array}{cc}1\hfill & \text{for}0<t<\pi \hfill \\ -1\hfill & \text{for}-\pi <t<0\hfill \\ c\hfill & \text{for}t=0,\pi \hfill \end{array}\right)b_k=2B_k=\frac{4}{\pi k}\text{for}\text{odd}k\ge 1$$
  and all other coefficients zero. Note that $c$ is any constant; the value of $c$ does not affect the coefficients $b_k$ . We have for $0<t<\pi$
  $$1=\frac{4}{\pi }(\text{sin}\left(t\right)+\frac{1}{3}\text{sin}\left(3t\right)+\frac{1}{5}\text{sin}\left(5t\right)+...)$$
  Note that for $t=0$ the value of the series on the right is 0, which is equal to $x(0-)+x(0+)$ , the middle of the jump of $x\left(t\right)$ at 0, no matter what $c$ is. Similar for $t=\pi$ .