### 4. Sampling and Aliasing

Overview: In chapter four, the conversion of signals between the analog and digital domains is studied. The basic ideas underlying sampling and signal reconstruction are presented. When sampling to convert a continuous-time (or analog) signal to a digital form for computer processing and storage, the primary issue is aliasing and the sampling strategy necessary to avoid aliasing of frequency components. The main objective of our presentation is an understanding of the Sampling Theorem which states that the sampling rate must be greater than twice the highest frequency contained in the analog signal. Frequency content is taken to mean the spectral content of a signal when represented as a sum of sinusoids.
The signal reconstruction of a D-to-A converter is presentated from a practical point of view as a generalization of interpolation.
Databases:

#### Demos - MATLAB6

Here are some movies that illustrate the concepts of aliasing and folding when a sinusoid is sampled below the Nyquist rate.
By visualizing the spectrogram of a synthesized chirp and listening to the sound, we experience the fact that a D-to-C converter cannot create output signals with frequencies higher than one half of the sampling frequency.
The Continuous-Discrete Sampling Demo (con2dis) is a program that shows the continuous and discrete spectra (and signals) during sampling.
Features:
• Users can change the input frequency and sampling rate.
• Frequency axis can be labeled in hertz or radians/sec.
• Reconstruction through D/A is also shown.
Here are some movies that illustrate the reconstruction process
These movies give an alternate view of the sampling process by using the strobing nature of a camcorder (30 frames per second) to show aliasing of a pattern on a rotating disk.
These movies were generated in MATLAB to show the strobe/sampling effect on a rotating disk. With MATLAB the rotation rate can be calibrated exactly, so that forward and backward movement of the spokes on the disk (due to aliasing) can be tracked.

#### Demos - LabVIEW1

The Continuous-Discrete Sampling Demo is a program that shows the continuous and discrete spectra (and signals) during sampling.

#### Labs - MATLAB2

The objective in this lab is to introduce digital images as a second useful signal type. We will show how the A-to-D sampling and the D-to-A reconstruction processes are carried out for digital images. In particular, we will show a commonly used method of image zooming (reconstruction) that gives poor results a later lab will revisit this issue and do a better job. [Files]
The objective of this lab is to study further the spectral content of signals analyzed via the spectrogram. There are several specific steps that will be considered in this lab:
1. Synthesize a linear-FM chirp with a Matlab M-file, and display its spectrogram. Choose the chirp parameters so that aliasing will happen.
2. Synthesize a periodic triangle wave with a Matlab M-file, and display its spectrogram. Relate the harmonic line spectrum to the fundamental period of the triangle wave.
3. Compare spectrograms using different scales for amplitude: decibels (dB) for amplitude versus linear amplitude.
4. Examine details of the harmonic lines in the dB spectrogram of the triangle wave.
5. Spectrogram: make a spectrogram of your voice signal, and relate the harmonic line spectrum to your previous measurement of pitch period.

#### Labs - LabVIEW1

The objective in this lab is to introduce digital images as a second useful signal type. We will show how the A-to-D sampling and the D-to-A reconstruction processes are carried out for digital images. In particular, we will show a commonly used method of image zooming (reconstruction) that gives poor results a later lab will revisit this issue and do a better job. [Files]