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.
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.
The Continuous-Discrete Sampling Demo is a program that shows the continuous
and discrete spectra (and signals) during sampling.
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:
- Synthesize a linear-FM chirp with a Matlab M-file, and display its spectrogram. Choose the chirp parameters so that aliasing will happen.
- 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.
- Compare spectrograms using different scales for amplitude: decibels (dB) for amplitude versus linear
amplitude.
- Examine details of the harmonic lines in the dB spectrogram of the triangle wave.
- Spectrogram: make a spectrogram of your voice signal, and relate the harmonic line spectrum to your
previous measurement of pitch period.
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]