Joseph Fourier discovered the Fourier transformation in the early 1800s. Fourier was a military scientist from France. The Fourier transform is a useful mathematical tool for obtaining the frequencies included in a time domain signal. Using the Fourier transform, we can rewrite every waveform as the sum of sine and cosine functions. It can be expressed mathematically as,

Message signal x(t) is multiplied by term exp(-j2*pi*f*t) or higher frequencies. And similarly, we recover the original signal at the receiver side by multiplying the exp(j2*pi*f*t) term. The integration sign denotes that the process is band limited. The above figure shows time is limited to –Î± to +Î± (where the periodic signal's frequency is not identically zero). However, while calculating bandwidth, we only consider positive components.

Let me show you a **real-world application of the Fourier transforms** in wireless communication. In the case of OFDM, we use the inverse fast Fourier transform, or IFFT at the TX side, to modulate the message signal with multicarrier frequencies. Then we use the fast Fourier transform, or FFT at the RX side, to retrieve the original signal at the receiver (which is the inverse of inverse fast Fourier transform at the transmitter side). The sum of sine and cosine signals can represent any periodic signal. So, first, we shift the original message's frequency to a higher frequency, then we prepare it as a summation of several sine and cosine signals using multiple carrier modulation. Then we retrieve the original signal at the receiver side using the inverse approach.

## Inverse Fourier Transform

**x(t)** = integration on interval [-infinite to infinite] **{X(f)exp(j2Ï€ft)}/2****Ï€**

Where, X(f) is the Fourier transform of x(t)

## Discrete Fourier Transform (DFT)

## Fast Fourier Transform (FFT)

## Parseval's Theorem:

According to Parseval's theorem, the Fourier Transform's Duality Property states that the total energy contained in all samples across time is equal to the spectral power across frequency. Using this property, we can determine power from a signal's spectrum which is far easier than calculating power from its time domain.

## Applications of Fourier Transform

## Fourier Transform Chapters

- Advantages of FFT over DFT and diagram of FFT Algorithm
- Properties of the Fourier Transform
- Power Spectral Density (PSD)