7.3 Fourier Series and Transforms

7.3 Fourier Series and Transforms

1. Continuous-Time (CT) Fourier Series for Periodic Signals

1.1 Representation of Periodic Signals

A periodic continuous-time signal with period $T_0$ satisfies: $x(t) = x(t + T_0) \quad \text{for all } t$

• Fundamental period: Smallest $T_0 > 0$ satisfying above.

• Fundamental frequency: $f_0 = \frac{1}{T_0}$ Hz, $\omega_0 = 2\pi f_0 = \frac{2\pi}{T_0}$ rad/s.

1.2 Trigonometric Fourier Series

1. General Form: $x(t) = a_0 + \sum_{n=1}^{\infty} [a_n \cos(n\omega_0 t) + b_n \sin(n\omega_0 t)]$

2. Coefficients Calculation:

   • DC component: $a_0 = \frac{1}{T_0} \int_{T_0} x(t) dt$

   • Cosine coefficients: $a_n = \frac{2}{T_0} \int_{T_0} x(t) \cos(n\omega_0 t) dt$

   • Sine coefficients: $b_n = \frac{2}{T_0} \int_{T_0} x(t) \sin(n\omega_0 t) dt$

3. Symmetry Properties:

   • Even signals: $b_n = 0$ for all $n$

   • Odd signals: $a_n = 0$ for all $n$ (including $a_0$)

1.3 Exponential (Complex) Fourier Series

1. General Form: $x(t) = \sum_{n=-\infty}^{\infty} c_n e^{jn\omega_0 t}$

2. Coefficients Calculation: $c_n = \frac{1}{T_0} \int_{T_0} x(t) e^{-jn\omega_0 t} dt$

3. Relationship with Trigonometric Coefficients: $c_0 = a_0$ $c_n = \frac{1}{2}(a_n - jb_n) \quad \text{for } n > 0$ $c_{-n} = \frac{1}{2}(a_n + jb_n) \quad \text{for } n > 0$ $a_n = c_n + c_{-n}$ $b_n = j(c_n - c_{-n})$

1.4 Properties of Fourier Series Coefficients

1. Linearity: If $z(t) = \alpha x(t) + \beta y(t)$, then $z_n = \alpha c_n^x + \beta c_n^y$

2. Time Shifting: If $y(t) = x(t - t_0)$, then $c_n^y = c_n^x e^{-jn\omega_0 t_0}$

3. Time Reversal: If $y(t) = x(-t)$, then $c_n^y = c_{-n}^x$

4. Time Scaling: If $y(t) = x(at)$, $a > 0$, period becomes $T_0/a$, $\omega_0' = a\omega_0$

5. Convolution: Periodic convolution leads to multiplication of coefficients.

6. Parseval's Theorem for Periodic Signals: $\frac{1}{T_0} \int_{T_0} |x(t)|^2 dt = \sum_{n=-\infty}^{\infty} |c_n|^2$

1.5 Dirichlet Conditions for Fourier Series Convergence

For Fourier series to converge to $x(t)$ at continuity points:

1. $x(t)$ is absolutely integrable over one period.

2. $x(t)$ has finite number of maxima/minima in one period.

3. $x(t)$ has finite number of discontinuities in one period.

4. At discontinuities, Fourier series converges to average of left and right limits.

2. Fourier Integral and Continuous-Time Fourier Transform (CTFT)

2.1 Fourier Transform for Aperiodic Signals

For aperiodic signals, we use Fourier Transform (extends Fourier series concept).

1. Forward Fourier Transform: $X(j\omega) = \mathcal{F}\{x(t)\} = \int_{-\infty}^{\infty} x(t) e^{-j\omega t} dt$

2. Inverse Fourier Transform: $x(t) = \mathcal{F}^{-1}\{X(j\omega)\} = \frac{1}{2\pi} \int_{-\infty}^{\infty} X(j\omega) e^{j\omega t} d\omega$

2.2 Existence Conditions (Dirichlet Conditions for CTFT)

For $X(j\omega)$ to exist:

1. $\int_{-\infty}^{\infty} |x(t)| dt < \infty$ (absolutely integrable)

2. Finite number of maxima/minima in any finite interval.

3. Finite number of finite discontinuities in any finite interval.

2.3 Relationship between Fourier Series and Fourier Transform

For periodic $x(t)$ with Fourier coefficients $c_n$: $X(j\omega) = 2\pi \sum_{n=-\infty}^{\infty} c_n \delta(\omega - n\omega_0)$ The Fourier transform of a periodic signal is a train of impulses at harmonic frequencies.

3. Key Properties of Fourier Transform

3.1 Symmetry Properties

1. Real-valued $x(t)$:

   • $X(-j\omega) = X^*(j\omega)$ (Conjugate symmetric)

   • Magnitude: $|X(-j\omega)| = |X(j\omega)|$ (even)

   • Phase: $\angle X(-j\omega) = -\angle X(j\omega)$ (odd)

2. Real and even $x(t)$:

   • $X(j\omega)$ is real and even.

3. Real and odd $x(t)$:

   • $X(j\omega)$ is purely imaginary and odd.

3.2 Basic Properties

1. Linearity: $\mathcal{F}\{\alpha x(t) + \beta y(t)\} = \alpha X(j\omega) + \beta Y(j\omega)$

2. Time Shifting: $\mathcal{F}\{x(t - t_0)\} = e^{-j\omega t_0} X(j\omega)$

3. Frequency Shifting: $\mathcal{F}\{e^{j\omega_0 t} x(t)\} = X(j(\omega - \omega_0))$

4. Time Scaling: $\mathcal{F}\{x(at)\} = \frac{1}{|a|} X\left(\frac{j\omega}{a}\right)$

5. Time Reversal: $\mathcal{F}\{x(-t)\} = X(-j\omega)$

6. Differentiation in Time: $\mathcal{F}\left\{\frac{d^n x(t)}{dt^n}\right\} = (j\omega)^n X(j\omega)$

7. Integration in Time: $\mathcal{F}\left\{\int_{-\infty}^{t} x(\tau) d\tau\right\} = \frac{1}{j\omega} X(j\omega) + \pi X(0) \delta(\omega)$

8. Differentiation in Frequency: $\mathcal{F}\{(-jt)^n x(t)\} = \frac{d^n}{d\omega^n} X(j\omega)$

9. Convolution in Time: $\mathcal{F}\{x(t) * h(t)\} = X(j\omega) H(j\omega)$

10. Multiplication in Time: $\mathcal{F}\{x(t) y(t)\} = \frac{1}{2\pi} X(j\omega) * Y(j\omega)$

3.3 Duality Property

If $\mathcal{F}\{x(t)\} = X(j\omega)$, then: $\mathcal{F}\{X(jt)\} = 2\pi x(-\omega)$

• Time domain ↔ Frequency domain symmetry.

4. Parseval's Theorem (Power/Energy Conservation)

4.1 For Energy Signals (Aperiodic)

Total energy in time domain = Total energy in frequency domain: $\int_{-\infty}^{\infty} |x(t)|^2 dt = \frac{1}{2\pi} \int_{-\infty}^{\infty} |X(j\omega)|^2 d\omega$

• $|X(j\omega)|^2$ is called the energy spectral density.

4.2 For Power Signals (Periodic)

Average power in time domain = Sum of squared Fourier coefficients: $\frac{1}{T_0} \int_{T_0} |x(t)|^2 dt = \sum_{n=-\infty}^{\infty} |c_n|^2$

4.3 Physical Interpretation

1. Conservation of energy/power between time and frequency domains.

2. Energy spectral density shows how signal energy is distributed across frequencies.

3. Basis for filter design and signal analysis.

5. Common Fourier Transform Pairs

5.1 Elementary Functions

1. Rectangular Pulse: $\text{rect}\left(\frac{t}{\tau}\right) \leftrightarrow \tau \text{sinc}\left(\frac{\omega\tau}{2}\right)$ where $\text{sinc}(x) = \frac{\sin x}{x}$

2. Unit Impulse: $\delta(t) \leftrightarrow 1$

3. Constant: $1 \leftrightarrow 2\pi \delta(\omega)$

4. Unit Step: $u(t) \leftrightarrow \frac{1}{j\omega} + \pi \delta(\omega)$

5.2 Exponential Functions

1. Decaying Exponential: $e^{-at} u(t) \leftrightarrow \frac{1}{a + j\omega}, \quad a > 0$

2. Two-sided Exponential: $e^{-a|t|} \leftrightarrow \frac{2a}{a^2 + \omega^2}, \quad a > 0$

5.3 Trigonometric Functions

1. Cosine: $\cos(\omega_0 t) \leftrightarrow \pi[\delta(\omega - \omega_0) + \delta(\omega + \omega_0)]$

2. Sine: $\sin(\omega_0 t) \leftrightarrow j\pi[\delta(\omega + \omega_0) - \delta(\omega - \omega_0)]$

5.4 Gaussian Pulse

$e^{-at^2} \leftrightarrow \sqrt{\frac{\pi}{a}} e^{-\frac{\omega^2}{4a}}, \quad a > 0$

6. Discrete-Time Fourier Series (DTFS)

6.1 Representation of Periodic Discrete-Time Signals

A periodic discrete-time signal with period $N$ satisfies: $x[n] = x[n + N] \quad \text{for all } n$

• Fundamental period: Smallest $N > 0$ satisfying above.

• Fundamental frequency: $\Omega_0 = \frac{2\pi}{N}$ rad/sample.

6.2 DTFS Representation

$x[n] = \sum_{k=\langle N\rangle} a_k e^{jk\Omega_0 n}$ where $\langle N\rangle$ denotes any $N$ consecutive integers (typically $0 \leq k \leq N-1$).

6.3 DTFS Coefficients

$a_k = \frac{1}{N} \sum_{n=\langle N\rangle} x[n] e^{-jk\Omega_0 n}$

6.4 Properties of DTFS

1. Periodicity of coefficients: $a_{k+N} = a_k$

2. Linearity: Same as CT Fourier series.

3. Time shifting: If $y[n] = x[n - n_0]$, then $a_k^y = a_k^x e^{-jk\Omega_0 n_0}$

4. Parseval's Theorem for DTFS: $\frac{1}{N} \sum_{n=\langle N\rangle} |x[n]|^2 = \sum_{k=\langle N\rangle} |a_k|^2$

7. Discrete-Time Fourier Transform (DTFT)

7.1 Definition of DTFT

For aperiodic discrete-time signals:

1. Forward DTFT: $X(e^{j\Omega}) = \sum_{n=-\infty}^{\infty} x[n] e^{-j\Omega n}$

2. Inverse DTFT: $x[n] = \frac{1}{2\pi} \int_{2\pi} X(e^{j\Omega}) e^{j\Omega n} d\Omega$

   • Integration over any $2\pi$ interval.

7.2 Properties of DTFT

1. Periodicity: $X(e^{j(\Omega + 2\pi)}) = X(e^{j\Omega})$ (periodic with period $2\pi$)

2. Linearity: $\mathcal{F}\{\alpha x[n] + \beta y[n]\} = \alpha X(e^{j\Omega}) + \beta Y(e^{j\Omega})$

3. Time Shifting: $\mathcal{F}\{x[n - n_0]\} = e^{-j\Omega n_0} X(e^{j\Omega})$

4. Frequency Shifting: $\mathcal{F}\{e^{j\Omega_0 n} x[n]\} = X(e^{j(\Omega - \Omega_0)})$

5. Time Reversal: $\mathcal{F}\{x[-n]\} = X(e^{-j\Omega})$

6. Differentiation in Frequency: $\mathcal{F}\{n x[n]\} = j \frac{d}{d\Omega} X(e^{j\Omega})$

7. Convolution: $\mathcal{F}\{x[n] * h[n]\} = X(e^{j\Omega}) H(e^{j\Omega})$

8. Multiplication: $\mathcal{F}\{x[n] y[n]\} = \frac{1}{2\pi} X(e^{j\Omega}) \circledast Y(e^{j\Omega})$ where $\circledast$ denotes periodic convolution.

7.3 Symmetry Properties (Real-valued $x[n]$)

1. $X(e^{-j\Omega}) = X^*(e^{j\Omega})$ (Conjugate symmetric)

2. $\text{Re}\{X(e^{j\Omega})\}$ is even: $\text{Re}\{X(e^{-j\Omega})\} = \text{Re}\{X(e^{j\Omega})\}$

3. $\text{Im}\{X(e^{j\Omega})\}$ is odd: $\text{Im}\{X(e^{-j\Omega})\} = -\text{Im}\{X(e^{j\Omega})\}$

4. $|X(e^{j\Omega})|$ is even: $|X(e^{-j\Omega})| = |X(e^{j\Omega})|$

5. $\angle X(e^{j\Omega})$ is odd: $\angle X(e^{-j\Omega}) = -\angle X(e^{j\Omega})$

7.4 Parseval's Theorem for DTFT

$\sum_{n=-\infty}^{\infty} |x[n]|^2 = \frac{1}{2\pi} \int_{2\pi} |X(e^{j\Omega})|^2 d\Omega$

• $|X(e^{j\Omega})|^2$ is the energy spectral density for discrete-time signals.

7.5 Common DTFT Pairs

1. Unit Impulse: $\delta[n] \leftrightarrow 1$

2. Unit Step: $u[n] \leftrightarrow \frac{1}{1 - e^{-j\Omega}} + \pi \sum_{k=-\infty}^{\infty} \delta(\Omega - 2\pi k)$

3. Exponential Sequence: $a^n u[n] \leftrightarrow \frac{1}{1 - ae^{-j\Omega}}, \quad |a| < 1$

4. Rectangular Pulse: $x[n] = \begin{cases} 1 & |n| \leq N \\ 0 & \text{otherwise} \end{cases}$ $\leftrightarrow \frac{\sin(\Omega(N+\frac{1}{2}))}{\sin(\Omega/2)}$

5. Ideal Low-pass Filter: $\frac{\sin(\Omega_c n)}{\pi n} \leftrightarrow \begin{cases} 1 & |\Omega| \leq \Omega_c \\ 0 & \Omega_c < |\Omega| \leq \pi \end{cases}$

7.6 Relationship between CTFT and DTFT

For sampled signal $x_s(t) = \sum_{n=-\infty}^{\infty} x[n] \delta(t - nT)$: $X_s(j\omega) = \frac{1}{T} \sum_{k=-\infty}^{\infty} X\left(j\left(\omega - \frac{2\pi k}{T}\right)\right)$ and $X(e^{j\Omega}) = X_s(j\omega)\big|_{\omega=\Omega/T}$

8. Important Formulas Summary

8.1 Fourier Series Summary

1. CT Fourier Series: $x(t) = \sum_{n=-\infty}^{\infty} c_n e^{jn\omega_0 t}, \quad c_n = \frac{1}{T_0} \int_{T_0} x(t) e^{-jn\omega_0 t} dt$

2. DT Fourier Series: $x[n] = \sum_{k=\langle N\rangle} a_k e^{jk\Omega_0 n}, \quad a_k = \frac{1}{N} \sum_{n=\langle N\rangle} x[n] e^{-jk\Omega_0 n}$

8.2 Fourier Transform Summary

1. CT Fourier Transform: $X(j\omega) = \int_{-\infty}^{\infty} x(t) e^{-j\omega t} dt, \quad x(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} X(j\omega) e^{j\omega t} d\omega$

2. DT Fourier Transform: $X(e^{j\Omega}) = \sum_{n=-\infty}^{\infty} x[n] e^{-j\Omega n}, \quad x[n] = \frac{1}{2\pi} \int_{2\pi} X(e^{j\Omega}) e^{j\Omega n} d\Omega$

8.3 Parseval's Theorems

1. CT Energy Signals: $\int_{-\infty}^{\infty} |x(t)|^2 dt = \frac{1}{2\pi} \int_{-\infty}^{\infty} |X(j\omega)|^2 d\omega$

2. DT Energy Signals: $\sum_{n=-\infty}^{\infty} |x[n]|^2 = \frac{1}{2\pi} \int_{2\pi} |X(e^{j\Omega})|^2 d\Omega$

3. CT Periodic Signals: $\frac{1}{T_0} \int_{T_0} |x(t)|^2 dt = \sum_{n=-\infty}^{\infty} |c_n|^2$

4. DT Periodic Signals: $\frac{1}{N} \sum_{n=\langle N\rangle} |x[n]|^2 = \sum_{k=\langle N\rangle} |a_k|^2$

8.4 Key Relationships

1. Periodicity: DTFT is periodic with period $2\pi$, CTFT is not periodic.

2. Frequency Variables:

   • CT: $\omega$ in rad/sec, $f$ in Hz

   • DT: $\Omega$ in rad/sample, $\Omega = \omega T$

3. Sampling Relationship: $X(e^{j\Omega}) = \frac{1}{T} \sum_{k=-\infty}^{\infty} X\left(j\frac{\Omega - 2\pi k}{T}\right)$

4. Convergence: Fourier series converges pointwise, Fourier transform converges in mean-square sense.