Summary of Digital Signal Processing Fundamentals

## Introduction Fourier analysis studies how signals can be represented as sums or integrals of sinusoids. It connects time-domain behavior of signals with their frequency content, allowing analysis, filtering, and synthesis in many engineering and physics applications. This material covers complex Fourier series, phasor interpretation, the rectangular signal example, spectrum behavior, Gibbs phenomenon, and the Fourier transform limit. > **Definition:** A sinusoidal signal can be represented by a rotating phasor, a complex exponential whose real part yields a cosine waveform. ## 1. Rotating Phasors and Sinusoids ### Complex phasor representation A sinusoid with amplitude $A$ and phase $\phi$ can be written as the real part of a complex exponential: $A\cos(\omega t + \phi) = \Re\left\{A e^{j\phi} e^{j\omega t}\right\}$. Each complex coefficient therefore encodes two real values: amplitude and phase. > **Definition:** A phasor is a complex number $C = |C|e^{j\arg(C)}$ representing amplitude $|C|$ and phase $\arg(C)$ of a sinusoid. ### Real-valued sinusoids from conjugate phasors A pair of complex conjugate phasors $C_k e^{j\omega_k t}$ and $C_{-k} e^{j\omega_{-k} t}$ with $C_{-k} = C_k^{*}$ combine to produce the real sinusoid $$s_k(t) = C_k e^{j\omega_k t} + C_k^{*} e^{-j\omega_k t} = 2\Re\{C_k e^{j\omega_k t}\} = 2|C_k|\cos\left(\omega_k t + \arg(C_k)\right).$$ ## 2. Complex Fourier Series (CFS) ### Formula for coefficients For a periodic signal $s(t)$ with period $T$, the complex Fourier series coefficient $\hat{c}_k$ is $$\hat{c}_k = \frac{1}{T}\int_{T} s(t) e^{-j 2\pi k f_1 t} \, dt$$ where $f_1 = \frac{1}{T}$ is the fundamental frequency and the integral is taken over any period of length $T$. > **Definition:** The complex Fourier series expresses a periodic signal as $s(t) = \sum_{k=-\infty}^{\infty} \hat{c}_k e^{j 2\pi k f_1 t}$. ### Inverse relation Given coefficients $\hat{c}_k$, reconstruct the signal by $$s(t) = \sum_{k=-\infty}^{\infty} \hat{c}_k e^{j 2\pi k f_1 t}.$$ ## 3. Example: Rectangular (Pulse) Signal Consider a periodic rectangular pulse of period $T$ and pulse width $\tau$ centered in each period and amplitude 1. The complex coefficient calculation yields a sinc-shaped spectrum. Calculate $$\hat{c}_k = \frac{1}{T} \int_{-\tau/2}^{\tau/2} 1 \cdot e^{-j 2\pi k f_1 t} \, dt = \frac{1}{T} \left[ \frac{e^{-j 2\pi k f_1 t}}{-j 2\pi k f_1} \right]_{-\tau/2}^{\tau/2}.$$ After simplification one obtains $$\hat{c}_k = \frac{\tau}{T} \operatorname{sinc}\left(2\pi k f_1 \frac{\tau}{2}\right)$$ where we define $$\operatorname{sinc}(x) = \frac{\sin(x)}{x}$$ and interpret the $k=0$ term with the limit giving $\hat{c}_0 = \frac{\tau}{T}$. > **Definition:** The sinc function is $\operatorname{sinc}(x) = \frac{\sin(x)}{x}$ with $\operatorname{sinc}(0) = 1$. ### Spectrum characteristics - The amplitude spectrum follows $|\hat{c}_k| \propto \operatorname{sinc}(\cdot)$, producing main lobe and decaying side lobes. - The phase spectrum is zero for a symmetric, centered pulse (coefficients are real and even). Fun fact: The discrete spectral lines of a periodic pulse collapse into a continuous sinc envelope as the period $T$ increases and the frequency spacing $f_1 = 1/T$ decreases, leading to the Fourier transform of a single pulse. ## 4. Partial Sums and Gibbs Phenomenon When reconstructing a discontinuous periodic signal from a finite number of harmonics, the partial sum overshoots at discontinuities. This is the Gibbs phenomenon: - Overshoot amplitude approaches about $11\%$ of the jump magnitude regardless of the number of harmonics. - Overshoot width decreases as more harmonics are included. Practical mitigation includes applying windowing (weight) functions to spectral coefficients before summation. ## 5. From Fourier Series to Fourier Transform ### Transition idea For a periodic signal with period $T$, spectral coefficients are at discrete frequencies $f_k = k f_1$ with spacing $f_1 = 1/T$. Normalize coef