The complex exponential power form of Fourier series of x(t) is

\(x\left( t \right) = \mathop \sum \limits_{k = - \infty }^\infty {a_k}{e^{j\frac{{2\pi }}{{{T_0}}}kt}}\)

If \(x\left( t \right) = \mathop \sum \limits_{b = - \infty }^\infty \delta \left( {t - b} \right)\), then the value of a_{k} is :

This question was previously asked in

ESE Electronics 2016 Paper 1: Official Paper

Option 3 : 1

__Concept:__

- The main aim of the Fourier series is one period to many frequencies.
- All
**non-sinusoidal**signals can be converted**into sinusoidal**frequency by this method. - It is applicable only to periodic signals.

This can be represented in two ways.

1.Trigonometric Fourier series (TFS)

2. Exponential Fourier series (EFS): it is also known as complex Fourier series.

Exponential Fourier series is represented as:

\(g\left( t \right) = \mathop \sum \limits_{n = - \infty }^\infty {C_n}{e^{jn\omega t}}\)

C_{n} is calculated as:

\({C_n} = \frac{1}{T}\mathop \smallint \limits_0^T g\left( t \right){e^{ - jn{\omega _0}t}}\;dt\)

C_{0 }= a_{0}

__Calculation:__

Given x(t) is

\(\mathop \sum \limits_{b = - \infty }^\infty \delta \left( {t - b} \right)\)

From the given Fourier series equation the Fourier coefficient is

\({a_k} = \frac{1}{T}\mathop \smallint \limits_0^T x\left( t \right){e^{ - j\frac{{2\pi }}{T}kt}}dt\)

From the spectrum, the period is T = 1 sec

\({a_k} = \frac{1}{T}\mathop \smallint \limits_0^1 \delta \left( t \right){e^{ - j\frac{{2\pi }}{1}kt}}dt\)

Impulse function only exists at t = 0 sec. So above integral is written as:

\({a_k} = \frac{1}{1}{e^{ - j\frac{{2\pi }}{1}k \times 0}}\mathop \smallint \limits_0^1 \delta \left( t \right)dt\)

**a _{k }= 1**

__Extra concept:__

Trigonometric Fourier series represented as:

\(g\left( t \right) = {a_0} + \mathop \sum \limits_{n = 1\;}^\infty {a_n}cosn{\omega _0}t + {b_n}sinn{\omega _0}t\)

Where a_{0}, a_{1}, a_{2} are calculated by

\({a_0} = \frac{1}{T}\mathop \smallint \limits_0^T g\left( t \right)dt = \frac{{Area\;of\;g\left( t \right)\;over\;one\;period\;}}{{Fundamental\;period}}\)

\({a_n} = \frac{2}{T}\mathop \smallint \limits_0^T g\left( t \right)cosn{\omega _0}t\;dt\)

\({b_n} = \frac{2}{T}\mathop \smallint \limits_0^T g\left( t \right)sinn{\omega _0}t\;dt\)

**Interconversion** between EFS and TFS

TFS to EFS |
EFS to TFS |

c |
a |

\({C_n} = \frac{{{a_n} - j{b_n}}}{2}\) \(\; = \frac{1}{T}\mathop \smallint \limits_0^T g\left( t \right){e^{ - jn{\omega _0}t}}dt\;\) |
a |

\({C_{ - n}} = \frac{{{a_n} + j{b_n}}}{2}\) |
jb |