Fourier Series - Continuous Time
Fourier Series - Continuous Time Periodic Signals
Definition
Any periodic signal \(x(t)\) with period \(T\) can be represented as:
Fourier Coefficients
Trigonometric Form of Fourier Series
Alternative Representation
Coefficients
Properties of Fourier Series
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Linearity: \(ax_1(t) + bx_2(t) \leftrightarrow aa_{1n} + ba_{2n}\)
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Time Shifting: \(x(t-t_0) \leftrightarrow a_n e^{-jn\omega_0 t_0}\)
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Frequency Shifting: \(x(t)e^{jm\omega_0 t} \leftrightarrow a_{n-m}\)
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Time Reversal: \(x(-t) \leftrightarrow a_{-n}\)
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Conjugation: \(x^*(t) \leftrightarrow a_{-n}^*\)
Parseval’s Theorem
Fourier Series - Discrete Time
Discrete-Time Fourier Series (DTFS)
Definition
For a discrete-time periodic signal \(x[n]\) with period \(N\):
Fourier Coefficients
Properties of DTFS
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Linearity: \(ax_1[n] + bx_2[n] \leftrightarrow aa_{1k} + ba_{2k}\)
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Time Shifting: \(x[n-n_0] \leftrightarrow a_k e^{-jk\omega_0 n_0}\)
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Frequency Shifting: \(x[n]e^{jm\omega_0 n} \leftrightarrow a_{k-m}\)
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Time Reversal: \(x[-n] \leftrightarrow a_{-k}\)
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Conjugation: \(x^*[n] \leftrightarrow a_{-k}^*\)
Parseval’s Theorem
Fourier Transform - Continuous Time
Continuous-Time Fourier Transform (CTFT)
Forward Transform
Inverse Transform
Existence Conditions
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Absolutely integrable: \(\int_{-\infty}^{\infty} |x(t)| dt < \infty\)
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Finite energy: \(\int_{-\infty}^{\infty} |x(t)|^2 dt < \infty\)
Common 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 \(a > 0\)) |
\(\text{rect}(t/T)\) | \(T\text{sinc}(\omega T/2)\) |
Properties of CTFT
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Linearity: \(ax_1(t) + bx_2(t) \leftrightarrow aX_1(\omega) + bX_2(\omega)\)
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Time Shifting: \(x(t-t_0) \leftrightarrow X(\omega)e^{-j\omega t_0}\)
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Frequency Shifting: \(x(t)e^{j\omega_0 t} \leftrightarrow X(\omega - \omega_0)\)
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Time Scaling: \(x(at) \leftrightarrow \frac{1}{|a|}X(\omega/a)\)
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Differentiation: \(\frac{dx(t)}{dt} \leftrightarrow j\omega X(\omega)\)
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Integration: \(\int_{-\infty}^{t} x(\tau) d\tau \leftrightarrow \frac{X(\omega)}{j\omega} + \pi X(0)\delta(\omega)\)
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Convolution: \(x_1(t) * x_2(t) \leftrightarrow X_1(\omega)X_2(\omega)\)
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Multiplication: \(x_1(t)x_2(t) \leftrightarrow \frac{1}{2\pi}X_1(\omega) * X_2(\omega)\)
Fourier Transform - Discrete Time
Discrete-Time Fourier Transform (DTFT)
Forward Transform
Inverse Transform
Properties
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\(X(\omega)\) is periodic with period \(2\pi\)
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\(X(\omega + 2\pi) = X(\omega)\)
Common 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 \(|a| < 1\)) |
\(\text{rect}_N[n]\) | \(\frac{\sin(\omega(N+1)/2)}{\sin(\omega/2)}e^{-j\omega N/2}\) |
Properties of DTFT
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Linearity: \(ax_1[n] + bx_2[n] \leftrightarrow aX_1(\omega) + bX_2(\omega)\)
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Time Shifting: \(x[n-n_0] \leftrightarrow X(\omega)e^{-j\omega n_0}\)
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Frequency Shifting: \(x[n]e^{j\omega_0 n} \leftrightarrow X(\omega - \omega_0)\)
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Time Reversal: \(x[-n] \leftrightarrow X(-\omega)\)
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Conjugation: \(x^*[n] \leftrightarrow X^*(-\omega)\)
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Convolution: \(x_1[n] * x_2[n] \leftrightarrow X_1(\omega)X_2(\omega)\)
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Multiplication: \(x_1[n]x_2[n] \leftrightarrow \frac{1}{2\pi}X_1(\omega) * X_2(\omega)\)
Applications
Applications of Fourier Transform
Signal Processing Applications
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Filtering: Design and analysis of filters
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Spectral Analysis: Frequency domain representation
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Modulation: AM, FM, and digital modulation schemes
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Sampling: Aliasing and anti-aliasing filters
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Communication Systems: Channel analysis and equalization
System Analysis
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Frequency Response: \(H(\omega) = \mathcal{F}\{h(t)\}\)
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System Stability: From pole-zero locations
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Bandwidth: System and signal bandwidth calculations
Energy and Power Spectral Density
Energy Spectral Density
For energy signals:
Power Spectral Density
For power signals: