GATE EE

Fourier Analysis in Signals & Systems : GATE Quick Notes

Lecture Notes

SEC 01

Fourier Series - Continuous Time

SEC 02

Fourier Series - Continuous Time Periodic Signals

1Fourier Series - Continuous Time Periodic Signals
1Definition

Any periodic signal \(x(t)\) with period \(T\) can be represented as:

\[\begin{aligned} x(t) &= \sum_{n=-\infty}^{\infty} a_n e^{jn\omega_0 t} \\ \text{where } \omega_0 &= \frac{2\pi}{T} \end{aligned}\]
1Fourier Coefficients
\[\begin{aligned} a_n &= \frac{1}{T} \int_{T} x(t) e^{-jn\omega_0 t} dt \end{aligned}\]
SEC 03

Trigonometric Form of Fourier Series

1Trigonometric Form of Fourier Series
1Alternative Representation
\[\begin{aligned} x(t) &= a_0 + \sum_{n=1}^{\infty} [a_n \cos(n\omega_0 t) + b_n \sin(n\omega_0 t)] \end{aligned}\]
1Coefficients
\[\begin{aligned} a_0 &= \frac{1}{T} \int_{T} x(t) dt \\ a_n &= \frac{2}{T} \int_{T} x(t) \cos(n\omega_0 t) dt \\ b_n &= \frac{2}{T} \int_{T} x(t) \sin(n\omega_0 t) dt \end{aligned}\]
SEC 04

Properties of Fourier Series

1Properties of Fourier Series

  • Linearity: \(ax_1(t) + bx_2(t) \leftrightarrow aa_{1n} + ba_{2n}\)

  • Time Shifting: \(x(t-t_0) \leftrightarrow a_n e^{-jn\omega_0 t_0}\)

  • Frequency Shifting: \(x(t)e^{jm\omega_0 t} \leftrightarrow a_{n-m}\)

  • Time Reversal: \(x(-t) \leftrightarrow a_{-n}\)

  • Conjugation: \(x^*(t) \leftrightarrow a_{-n}^*\)

1Parseval’s Theorem
\[\begin{aligned} \frac{1}{T} \int_{T} |x(t)|^2 dt = \sum_{n=-\infty}^{\infty} |a_n|^2 \end{aligned}\]
SEC 05

Fourier Series - Discrete Time

SEC 06

Discrete-Time Fourier Series (DTFS)

1Discrete-Time Fourier Series (DTFS)
1Definition

For a discrete-time periodic signal \(x[n]\) with period \(N\):

\[\begin{aligned} x[n] &= \sum_{k=0}^{N-1} a_k e^{jk\omega_0 n} \\ \text{where } \omega_0 &= \frac{2\pi}{N} \end{aligned}\]
1Fourier Coefficients
\[\begin{aligned} a_k &= \frac{1}{N} \sum_{n=0}^{N-1} x[n] e^{-jk\omega_0 n} \end{aligned}\]
SEC 07

Properties of DTFS

1Properties of DTFS

  • Linearity: \(ax_1[n] + bx_2[n] \leftrightarrow aa_{1k} + ba_{2k}\)

  • Time Shifting: \(x[n-n_0] \leftrightarrow a_k e^{-jk\omega_0 n_0}\)

  • Frequency Shifting: \(x[n]e^{jm\omega_0 n} \leftrightarrow a_{k-m}\)

  • Time Reversal: \(x[-n] \leftrightarrow a_{-k}\)

  • Conjugation: \(x^*[n] \leftrightarrow a_{-k}^*\)

1Parseval’s Theorem
\[\begin{aligned} \frac{1}{N} \sum_{n=0}^{N-1} |x[n]|^2 = \sum_{k=0}^{N-1} |a_k|^2 \end{aligned}\]
SEC 08

Fourier Transform - Continuous Time

SEC 09

Continuous-Time Fourier Transform (CTFT)

1Continuous-Time Fourier Transform (CTFT)
1Forward Transform
\[\begin{aligned} X(\omega) = \mathcal{F}\{x(t)\} = \int_{-\infty}^{\infty} x(t) e^{-j\omega t} dt \end{aligned}\]
1Inverse Transform
\[\begin{aligned} x(t) = \mathcal{F}^{-1}\{X(\omega)\} = \frac{1}{2\pi} \int_{-\infty}^{\infty} X(\omega) e^{j\omega t} d\omega \end{aligned}\]
1Existence Conditions

  • Absolutely integrable: \(\int_{-\infty}^{\infty} |x(t)| dt < \infty\)

  • Finite energy: \(\int_{-\infty}^{\infty} |x(t)|^2 dt < \infty\)

SEC 10

Common CTFT Pairs

1Common CTFT Pairs
Signal Fourier Transform
\(\delta(t)\) \(1\)
\(1\) \(2\pi\delta(\omega)\)
\(e^{j\omega_0 t}\) \(2\pi\delta(\omega - \omega_0)\)
\(\cos(\omega_0 t)\) \(\pi[\delta(\omega - \omega_0) + \delta(\omega + \omega_0)]\)
\(\sin(\omega_0 t)\) \(j\pi[\delta(\omega + \omega_0) - \delta(\omega - \omega_0)]\)
\(e^{-at}u(t)\) \(\frac{1}{a + j\omega}\) (for \(a > 0\))
\(\text{rect}(t/T)\) \(T\text{sinc}(\omega T/2)\)
SEC 11

Properties of CTFT

1Properties of CTFT

  • Linearity: \(ax_1(t) + bx_2(t) \leftrightarrow aX_1(\omega) + bX_2(\omega)\)

  • Time Shifting: \(x(t-t_0) \leftrightarrow X(\omega)e^{-j\omega t_0}\)

  • Frequency Shifting: \(x(t)e^{j\omega_0 t} \leftrightarrow X(\omega - \omega_0)\)

  • Time Scaling: \(x(at) \leftrightarrow \frac{1}{|a|}X(\omega/a)\)

  • Differentiation: \(\frac{dx(t)}{dt} \leftrightarrow j\omega X(\omega)\)

  • Integration: \(\int_{-\infty}^{t} x(\tau) d\tau \leftrightarrow \frac{X(\omega)}{j\omega} + \pi X(0)\delta(\omega)\)

  • Convolution: \(x_1(t) * x_2(t) \leftrightarrow X_1(\omega)X_2(\omega)\)

  • Multiplication: \(x_1(t)x_2(t) \leftrightarrow \frac{1}{2\pi}X_1(\omega) * X_2(\omega)\)

SEC 12

Fourier Transform - Discrete Time

SEC 13

Discrete-Time Fourier Transform (DTFT)

1Discrete-Time Fourier Transform (DTFT)
1Forward Transform
\[\begin{aligned} X(\omega) = \mathcal{F}\{x[n]\} = \sum_{n=-\infty}^{\infty} x[n] e^{-j\omega n} \end{aligned}\]
1Inverse Transform
\[\begin{aligned} x[n] = \mathcal{F}^{-1}\{X(\omega)\} = \frac{1}{2\pi} \int_{-\pi}^{\pi} X(\omega) e^{j\omega n} d\omega \end{aligned}\]
1Properties

  • \(X(\omega)\) is periodic with period \(2\pi\)

  • \(X(\omega + 2\pi) = X(\omega)\)

SEC 14

Common DTFT Pairs

1Common DTFT Pairs
Signal Fourier Transform
\(\delta[n]\) \(1\)
\(1\) \(2\pi\sum_{k=-\infty}^{\infty}\delta(\omega - 2\pi k)\)
\(e^{j\omega_0 n}\) \(2\pi\sum_{k=-\infty}^{\infty}\delta(\omega - \omega_0 - 2\pi k)\)
\(\cos(\omega_0 n)\) \(\pi\sum_{k=-\infty}^{\infty}[\delta(\omega - \omega_0 - 2\pi k) + \delta(\omega + \omega_0 - 2\pi k)]\)
\(a^n u[n]\) \(\frac{1}{1 - ae^{-j\omega}}\) (for \(|a| < 1\))
\(\text{rect}_N[n]\) \(\frac{\sin(\omega(N+1)/2)}{\sin(\omega/2)}e^{-j\omega N/2}\)
SEC 15

Properties of DTFT

1Properties of DTFT

  • Linearity: \(ax_1[n] + bx_2[n] \leftrightarrow aX_1(\omega) + bX_2(\omega)\)

  • Time Shifting: \(x[n-n_0] \leftrightarrow X(\omega)e^{-j\omega n_0}\)

  • Frequency Shifting: \(x[n]e^{j\omega_0 n} \leftrightarrow X(\omega - \omega_0)\)

  • Time Reversal: \(x[-n] \leftrightarrow X(-\omega)\)

  • Conjugation: \(x^*[n] \leftrightarrow X^*(-\omega)\)

  • Convolution: \(x_1[n] * x_2[n] \leftrightarrow X_1(\omega)X_2(\omega)\)

  • Multiplication: \(x_1[n]x_2[n] \leftrightarrow \frac{1}{2\pi}X_1(\omega) * X_2(\omega)\)

SEC 16

Applications

SEC 17

Applications of Fourier Transform

1Applications of Fourier Transform
1Signal Processing Applications

  • Filtering: Design and analysis of filters

  • Spectral Analysis: Frequency domain representation

  • Modulation: AM, FM, and digital modulation schemes

  • Sampling: Aliasing and anti-aliasing filters

  • Communication Systems: Channel analysis and equalization

1System Analysis

  • Frequency Response: \(H(\omega) = \mathcal{F}\{h(t)\}\)

  • System Stability: From pole-zero locations

  • Bandwidth: System and signal bandwidth calculations

SEC 18

Energy and Power Spectral Density

1Energy and Power Spectral Density
1Energy Spectral Density

For energy signals:

\[\begin{aligned} S_{xx}(\omega) = |X(\omega)|^2 \end{aligned}\]
Total energy: \(E = \frac{1}{2\pi}\int_{-\infty}^{\infty} S_{xx}(\omega) d\omega\)
1Power Spectral Density

For power signals:

\[\begin{aligned} S_{xx}(\omega) = \lim_{T \to \infty} \frac{1}{T}|X_T(\omega)|^2 \end{aligned}\]
Average power: \(P = \frac{1}{2\pi}\int_{-\infty}^{\infty} S_{xx}(\omega) d\omega\)
SEC 19

Important Formulas - Summary

1Important Formulas - Summary
1Key Equations to Remember
\[\begin{aligned} \text{CTFS: } & x(t) = \sum_{n=-\infty}^{\infty} a_n e^{jn\omega_0 t} \\ \text{DTFS: } & x[n] = \sum_{k=0}^{N-1} a_k e^{jk\omega_0 n} \\ \text{CTFT: } & X(\omega) = \int_{-\infty}^{\infty} x(t) e^{-j\omega t} dt \\ \text{DTFT: } & X(\omega) = \sum_{n=-\infty}^{\infty} x[n] e^{-j\omega n} \\ \text{Parseval: } & \int |x(t)|^2 dt = \frac{1}{2\pi}\int |X(\omega)|^2 d\omega \end{aligned}\]