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.
Continuous Time Fourier Transform (CTFT)
Fourier transform is a process to convert a spatial domain signal (i.e., time domain signal) into a frequency domain signal. Oppositely, the inverse Fourier transform is a process to convert the frequency domain signal to the primary time domain signal.
Notation of CTFT
Let x(t) be a continuous-time signal. Then the CTFT is defined as:
\( X(j\omega) = \int_{-\infty}^{\infty} x(t) \cdot e^{-j\omega t} \, dt \)
Where:
- \( \omega \) is the angular frequency in radians/second.
- \( X(j\omega) \) is the frequency-domain representation of \( x(t) \).
- The transform assumes signals are absolutely integrable over time.
Inverse CTFT:
To reconstruct x(t) from its CTFT:
\( x(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} X(j\omega) \cdot e^{j\omega t} \, d\omega \)
Discrete Time Fourier Transform (DTFT)
The Discrete-Time Fourier Transform (DTFT) is used to analyze discrete-time signals, i.e., signals that are defined only at discrete intervals of time (like samples from an analog signal). These arise naturally in digital signal processing because all digital devices (computers, DSPs) process data in discrete form.
Notation of DTFT
Let x[n] be a discrete-time signal. Then the DTFT is defined as:
\( X(e^{j\omega}) = \sum_{n=-\infty}^{\infty} x[n] \cdot e^{-j\omega n} \)
Where:
- \( \omega \) is the angular frequency in radians/sample.
- \( e^{j\omega} \) represents the frequency-domain variable on the unit circle.
- \( X(e^{j\omega}) \) is periodic with period \( 2\pi \).
Inverse DTFT:
To reconstruct x[n] from its DTFT:
\( x[n] = \frac{1}{2\pi} \int_{-\pi}^{\pi} X(e^{j\omega}) \cdot e^{j\omega n} d\omega \)
Properties of Continuous Time Fourier Transform (CTFT)
Linearity
The Fourier Transform satisfies the property of linearity (superposition).
Consider two signals \( x_1(t) \) and \( x_2(t) \) with Fourier Transforms:
\( \mathcal{F}\{x_1(t)\} = X_1(j\omega), \quad \mathcal{F}\{x_2(t)\} = X_2(j\omega) \)
Then for any constants \( a_1 \) and \( a_2 \), we have:
\( \mathcal{F}\{a_1 x_1(t) + a_2 x_2(t)\} = a_1 X_1(j\omega) + a_2 X_2(j\omega) \)
Scaling
If \( \mathcal{F}\{x(t)\} = X(j\omega) \), and \( a \) is a real constant, then:
\( \mathcal{F}\{x(at)\} = \frac{1}{|a|} X\left(\frac{j\omega}{a}\right) \)
Symmetry
If \( x(t) \) is real and even, then the Fourier Transform satisfies:
\( X(j\omega) = X^*(-j\omega) \)
If \( x(t) \) is real and odd, then:
\( X(j\omega) = -X^*(-j\omega) \)
Convolution
Fourier Transform converts the convolution of two signals in time domain into the multiplication of their transforms in frequency domain.
Time Domain Convolution
If \( F(x_1(t)) = X_1(\omega) \) and \( F(x_2(t)) = X_2(\omega) \), then:
\( F(x_1(t) * x_2(t)) = X_1(\omega) \cdot X_2(\omega) \quad \text{(‘*’ denotes convolution)} \)
Frequency Domain Convolution
If \( F(x_1(t)) = X_1(\omega) \), \( F(x_2(t)) = X_2(\omega) \), then:
\( F(x_1(t) \cdot x_2(t)) = \frac{1}{2\pi} X_1(\omega) * X_2(\omega) \quad \text{(‘*’ denotes convolution)} \)
Shifting Property
\( \mathcal{F}\{x(t - t_0)\} = e^{-j\omega t_0} X(\omega) \)
As a consequence, time shifting affects only the phase, leaving the magnitude spectrum \( |X(\omega)|^2 \) unchanged.
Duality
Duality states that if \( x(t) \leftrightarrow X(\omega) \), then the roles of time and frequency can be interchanged.
\( \mathcal{F}\{X(t)\} = 2\pi x(-\omega) \)
Differentiation
The Fourier Transform of the derivative of a signal corresponds to multiplication by \( j\omega \) in the frequency domain:
\( \mathcal{F} \left\{ \frac{d}{dt}x(t) \right\} = j\omega X(\omega) \)
Integration
Integration in the time domain corresponds to division by \( j\omega \) in the frequency domain:
\( \mathcal{F}\left\{\int_{-\infty}^{t} x(\tau) \, d\tau \right\} = \frac{X(\omega)}{j\omega} \)
Time Reversal
If \( x(t) \leftrightarrow X(\omega) \), then the Fourier Transform of \( x(-t) \) is \( X(-\omega) \). This means that time reversal corresponds to the reversal of the frequency spectrum in the frequency domain.
Applications of Fourier Transform
Fourier Transform Chapters
- Advantages of FFT over DFT and diagram of FFT Algorithm
- Properties of the Fourier Transform
- Fourier Transform of Dirac Delta Function Using MATLAB
- Power Spectral Density (PSD)
