Sampling

• Converts a continuous-time signal to discrete-time signal
• The signal is periodically measured over discrete intervals of time
  • Each interval is called a time step
• Can be quantized over a finite number of bits to obtain a digital signal
  • Bit depth is the number of bits used to represent a sample. This means that the range of the amplitude will be split into $2^n$ levels
  • SQNR measures the quantization precision
• Conditions for a signal to be unambiguously represented by its samples
  • Must be bandlimited
  • Sampled at Nyquist rate or higher

Bandlimited Signals

• Have a finite spectrum
  • There is an $f_\text{max} = \omega_{b}$ for which $X(j\omega) = 0$ for $|\omega| \gt \omega_{b}$
• Square waves are not bandlimited because they are represented by an infinite Fourier Series

Sampling and Spectrum

• Sampling a continuous-time signal creates a mapping between the spectra and the discrete-time domain with $\hat{\omega} = \frac{\omega}{f_s}$
  • This constrains the discrete-time frequencies to the interval $[-\pi, \pi]$ (known as the Nyquist interval), with the amplitudes remaining unchanged
• $\frac{\omega_b}{\pi} = f_s$, assuming $\omega_b$ in rad/s and $f_s$ in Hz
  • Basically, just remove $\pi$ to obtain the Nyquist rate from $\omega_b$
• discrete-time

References

• Sampling (course handout)
• ▶ Watch on YouTube