Training Fourier Analysis Signal Processing: Sampling, Filtering & Modulation
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Signal Processing: Sampling, Filtering & Modulation

35 min Fourier Analysis

Signal Processing: Sampling, Filtering & Modulation

The Nyquist-Shannon sampling theorem states that a bandlimited signal (zero energy above $B$ Hz) is perfectly reconstructed from samples at rate $>2B$ samples/second. Sampling at rate $f_s$ creates spectral copies (aliasing) separated by $f_s$; sufficiently high $f_s$ prevents overlap. Digital filters (FIR/IIR) implement frequency-selective operations in the DFT domain. AM/FM modulation translates signals in frequency to carrier frequencies for transmission.

Sampling Theorem

A signal $f(t)$ bandlimited to $|\xi|B$) is completely determined by its samples $\{f(n/f_s)\}_{n\in\mathbb{Z}}$ when $f_s>2B$ (Nyquist rate). Reconstruction: $f(t)=\sum_n f(n/f_s)\text{sinc}(f_s t-n)$ where $\text{sinc}(x)=\sin(\pi x)/(\pi x)$. Aliasing: if $f_s<2B$, high frequencies fold over — $f_s-B$ and $B$ become indistinguishable. Solution: anti-aliasing filter (low-pass at $f_s/2$) before sampling.

Ideal Filters & Filter Design

An ideal low-pass filter has transfer function $H(\xi)=1$ for $|\xi|

Example 1

A 44.1 kHz CD samples at $f_s=44100$ Hz. What is the highest frequency it can represent? Why does it capture human hearing?

Solution: Nyquist: maximum frequency $=f_s/2=22050$ Hz. Human hearing range: 20 Hz–20 kHz. Since $22050>20000$, the CD can represent all audible frequencies without aliasing. An anti-aliasing filter at $\ approx 20$ kHz is applied before ADC to suppress ultrasonic content.

Example 2

Design a simple 3-tap moving average FIR filter and find its frequency response.

Solution: $h=[1/3,1/3,1/3]$. $H(e^{j\omega})=\frac{1}{3}(1+e^{-j\omega}+e^{-2j\omega})=\frac{1}{3}e^{-j\omega}(e^{j\omega}+1+e^{-j\omega})=\frac{1}{3}e^{-j\omega}(1+2\cos\omega)$. Magnitude: $|H(\omega)|=\frac{|1+2\cos\omega|}{3}$. At $\omega=0$ (DC): $|H|=1$ (passes). At $\omega=\pi$ (Nyquist): $|H|=|1-2|/3=1/3$ (attenuates high freq). Low-pass behavior, cutoff $\approx 2\pi/3$.

Practice

  1. Prove the Shannon sampling theorem using the Poisson summation formula.
  2. Design a 4th-order Butterworth low-pass filter with cutoff $\omega_c=1$ rad/s and compute its transfer function $H(s)$.
  3. Explain FM modulation: if $f(t)$ is the message signal and $x(t)=A\cos(2\pi f_c t+2\pi k_f\int f\,dt)$, find the instantaneous frequency.
  4. Describe the z-transform and its relationship to the discrete-time Fourier transform for filter analysis.
Show Answer Key

1. By Poisson summation, $\sum_n f(n)=\sum_k\hat{f}(k)$. If $f$ is bandlimited to $[-B,B]$, then $\hat{f}$ is supported on $[-B,B]$. Sampling at rate $\ge2B$ means the periodized spectrum doesn't overlap, so $f$ is recoverable: $f(t)=\sum_n f(n/(2B))\text{sinc}(2Bt-n)$.

2. 4th-order Butterworth: $|H(j\omega)|^2=\frac{1}{1+\omega^8}$. The 4 poles in the left half-plane are at $s_k=e^{j\pi(2k+5)/8}$ for $k=0,1,2,3$. $H(s)=\frac{1}{(s^2+2\cos(5\pi/8)s+1)(s^2+2\cos(7\pi/8)s+1)}$.

3. Instantaneous frequency $f_i(t)=f_c+k_f f(t)$. The frequency deviation from the carrier is proportional to the message signal amplitude at each instant.

4. The $z$-transform of $x[n]$ is $X(z)=\sum_n x[n]z^{-n}$. Evaluating on the unit circle $z=e^{j\omega}$ gives the DTFT: $X(e^{j\omega})=\sum_n x[n]e^{-j\omega n}$. Filter analysis uses $H(z)$ (rational function) whose poles/zeros determine frequency response and stability.

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