Chapter Summary

Chapter 4 Summary: Signals and Systems

Key Points

• 1.

  CT signals are classified as energy or power signals. LTI systems are characterised by their impulse response $h(t)$; the output is the convolution $y = x * h$.

• 2.

  The Fourier transform decomposes signals into frequency components: $X(f) = \mathcal{F}\{x(t)\}$. Key properties: time shift $\to$ phase shift, modulation $\to$ frequency shift, convolution $\to$ multiplication.

• 3.

  The sampling theorem states that a band-limited signal ($W$ Hz) can be perfectly reconstructed from samples at rate $f_s > 2W$. Below this rate, aliasing occurs irreversibly.

• 4.

  Complex baseband representation strips the carrier from bandpass signals: $x(t) = \mathrm{Re}[\tilde{x}(t) e^{j2\pi f_c t}]$. All wireless analysis is done at baseband using I/Q components.

• 5.

  LTV systems generalise LTI with time-variant impulse response $h(t, \tau)$. The wireless channel is LTV; the quasi-static approximation treats it as piecewise LTI.

• 6.

  For WSS random processes through LTI systems: $S_Y(f) = |H(f)|^2 S_X(f)$. The matched filter $h(t) = s^*(T-t)$ maximises output SNR to $2E_s/N_0$.