Chapter 5: Sampling

Introduction

Sampling is the bridge between continuous-time and discrete-time signals. Understanding sampling theory is essential for digital signal processing.

Ideal Sampling

Definition: Ideal Sampling

Ideal sampling multiplies a continuous signal $x_c(t)$ by an impulse train:

$$x_s(t)=x_c(t)\sum _{n=-\mathrm{\infty }}^{+\mathrm{\infty }}\delta (t-n{T}_s)=\sum _{n=-\mathrm{\infty }}^{+\mathrm{\infty }}{x}_c(n{T}_s)\delta (t-n{T}_s)$$

where $T_s$ is the sampling period and $f_s=1/T_s$ is the sampling frequency.

Spectrum of Sampled Signal

Theorem: Frequency Domain Effect of Sampling

If $x_c(t)\leftrightarrow X_c(j\mathrm{\Omega })$, then the Fourier transform of the sampled signal is:

$$X_s(j\mathrm{\Omega })=\frac{1}{T_s}\sum _{k=-\mathrm{\infty }}^{+\mathrm{\infty }}{X}_c(j(\mathrm{\Omega }-k{\mathrm{\Omega }}_s))$$

where ${\mathrm{\Omega }}_s=2\pi f_s=2\pi /T_s$ is the sampling angular frequency.

Proposition: Spectral Replication

Sampling creates infinite replicas of the original spectrum, centered at multiples of the sampling frequency:

• Original spectrum centered at $\mathrm{\Omega }=0$
• Replicas centered at $\mathrm{\Omega }=\pm {\mathrm{\Omega }}_s,\pm 2{\mathrm{\Omega }}_s,\dots$

Nyquist-Shannon Sampling Theorem

Theorem: Nyquist-Shannon Sampling Theorem

A bandlimited signal $x_c(t)$ with maximum frequency $f_{max}$ (i.e., $X_c(j\mathrm{\Omega })=0$ for $|\mathrm{\Omega }|>2\pi f_{max}$) can be perfectly reconstructed from its samples if:

$$f_s>2f_{max}$$

or equivalently:

$$T_s<\frac{1}{2f_{max}}$$

The minimum sampling rate $f_N=2f_{max}$ is called the Nyquist rate.

Aliasing

Definition: Aliasing

Aliasing occurs when the sampling rate is too low ($f_s<2f_{max}$), causing spectral replicas to overlap. This results in distortion that cannot be corrected.

Example: Aliased Sinusoid

Consider sampling $x_c(t)=\cos (2\pi f_{0}t)$ at rate $f_s$.

If $f_0>f_s/2$, the sampled signal appears as:

$$x[n]=\cos (2\pi f_{a}n/f_s)$$

where $f_a=|f_0-k{f}_s|$ is the aliased frequency (for some integer $k$ that minimizes $f_a$).

For example: $f_0=900$ Hz sampled at $f_s=1000$ Hz gives $f_a=100$ Hz.

Normalized Frequency

Definition: Normalized Frequencies

For a continuous signal with frequency $f$ (Hz) or $\mathrm{\Omega }$ (rad/s):

Digital angular frequency:

$$\omega =\mathrm{\Omega }{T}_s=2\pi f/f_s\text{(rad/sample)}$$

Normalized frequency:

$$\nu =f/f_s\text{(cycles/sample)}$$

The unambiguous range is:

• $\omega \in [-\pi ,\pi ]$ or $[0,2\pi ]$
• $\nu \in [-0.5,0.5]$ or $[0,1]$

Ideal Reconstruction

Definition: Ideal Interpolation

Perfect reconstruction uses an ideal lowpass filter with cutoff $f_s/2$:

$$x_r(t)=\sum _{n=-\mathrm{\infty }}^{+\mathrm{\infty }}x[n]\text{sinc}\left(\frac{t-n{T}_s}{T_s}\right)$$

where $\text{sinc}(u)=\frac{\sin (\pi u)}{\pi u}$.

Theorem: Interpolation Formula

If the Nyquist criterion is satisfied, the original signal can be recovered:

$$x_c(t)=\sum _{n=-\mathrm{\infty }}^{+\mathrm{\infty }}{x}_c(n{T}_s)\text{sinc}\left(\frac{t-n{T}_s}{T_s}\right)$$

Practical Considerations

Anti-Aliasing Filter

Definition: Anti-Aliasing Filter

An anti-aliasing filter is a lowpass filter applied before sampling to remove frequencies above $f_s/2$:

$$|H_{aa}(j\mathrm{\Omega })|\approx \{\begin{array}{ll}1& |\mathrm{\Omega }|<\pi /T_s\\ 0& |\mathrm{\Omega }|>\pi /T_s\end{array}$$

Practical Reconstruction

Proposition: Zero-Order Hold (ZOH)

Practical DACs use a zero-order hold:

$$x_{zoh}(t)=x[n]\text{for }n{T}_s\le t<(n+1)T_s$$

The ZOH frequency response is:

$$H_{zoh}(j\mathrm{\Omega })=T_s\text{sinc}\left(\frac{\mathrm{\Omega }{T}_s}{2\pi }\right)e^{-j\mathrm{\Omega }{T}_s/2}$$

Oversampling and Decimation

Definition: Oversampling

Oversampling uses a sampling rate higher than the Nyquist rate: $f_s\gg 2f_{max}$.

Benefits:

• Relaxes anti-aliasing filter requirements
• Spreads quantization noise over wider bandwidth
• Enables noise shaping in sigma-delta converters

Definition: Decimation

Decimation by factor $M$ reduces the sampling rate:

$$y[n]=x[nM]$$

Must be preceded by lowpass filtering to prevent aliasing.

Definition: Interpolation (Upsampling)

Interpolation by factor $L$ increases the sampling rate:

$$y[n]=\{\begin{array}{ll}x[n/L]& n=0,\pm L,\pm 2L,\dots \\ 0& \text{otherwise}\end{array}$$

Must be followed by lowpass filtering to remove spectral images.