ELEC 241 Lab

Experiment 6.1

Sampling and Quantization

Part 1: Sample Rate and Aliasing

When we used the spectrum analyzer last week, the plots (of both the time function and the spectrum) were drawn with connected lines, so we could pretend that they were continuous functions. In fact they were not.

When we convert a continuous, analog signal to a digital signal (digitize it), we sample its value at regular intervals. The sequence of numbers that results represents the original signal at these sample points, but ignores what goes on between them. If the signal is sufficiently well-behaved (i.e. it satisfies the Nyquist criterion and contains no energy at frequencies greater than half the sampling frequency), then these sample points are enough to represent the original signal exactly. But if the original signal contains a frequency greater than half the sampling rate, that frequency will be aliased to a lower frequency.

Let's start by looking at what sampling looks like in the time domain.


Step 1:

 Connect the function generator output and CH1 of the scope to A/D input4 (pin 46 on the interface board socket strip).

Step 2:

 Set the function generator to produce a 5 V p-p, 300 Hz sine wave.

Step 3:

 Load the "wave display 1" Labview program. Note the sampling rate. Start the program by pressing the run button or by pressing CTRL-R with the cursor over the window.

Here's what we have:

\includegraphics[scale=0.650000]{ckt6.1.ps}


Step 4:

 You should see about three cycles of a sine wave displayed in the waveform graph. Unlike last week's display, the samples are shown as individual dots, rather than connected line segments.

As we increase the frequency of the sine wave, we get fewer samples of each cycle and the picture becomes less clear.

Step 5:

 Increase the frequency to about 2 kHz. Freeze the display by pressing the STOP button. You should see either several lines or several overlapping sine waves. This is an illusion caused by the fact that only a few samples of each cycle are being taken.

Question 1:

 Make a sketch to illustrate what happens when the frequency of the waveform is equal to (or nearly equal to) a small submultiple (e.g. 1/4) of the sampling rate.

Step 6:

 To see the actual underlying waveform more clearly, switch the display to connected lines: Place the cursor on the box marked "display style" underneath the display and select the continuous line style from the "Common Plots" submenu.

Step 7:

 Restart the program and continue increasing the frequency of the function generator until you reach 5 kHz, stopping at several points along the way to examine the waveform.

When the function generator frequency is exactly half the sampling frequency the samples will alternate the same positive and negative values. (This may be easier to see by switching to the vertical line plot display style.)

Continue increasing the function generator frequency. Notice that as you approach 10 kHz, you begin to see a well defined sine wave which decreases in frequency as you increase the function generator frequency. This is the alias of the generator frequency. At exactly 10 kHz you should get a zero frequency sine wave (it may help to turn the trigger on the waveform display to Off).

Step 8:

 Continue increasing the function generator frequency past 10 kHz. Note that you once again have a sine wave that increases in frequency as the input frequency increases.

Question 2:

 Derive an expression that relates the frequency of the sine wave displayed by the waveform display program to the actual frequency of the input sine wave.

Step 9:

 Try square and triangle waves of various frequencies and see what happens to them as the frequency changes.

Step 10:

 Press the STOP button and exit the waveform view program.

Part 2: Aliasing in the Frequency Domain



Step 1:

 Set the function generator to produce a 500 Hz, 5 V p-p sine wave.

Step 2:

 Load the "Spectrum Analyzer 5" program. This is similar to the program we used last week, but has a much lower sampling rate (10 kHz instead of 100 kHz). It also has an additional feature we'll examine in a few minutes.

We have a new circuit without having to move any wires. Here it is:

\includegraphics[scale=0.650000]{ckt6.2.ps}


Step 3:

 Start the spectrum analyzer program. You should see a display similar to last week's, but with a higher noise level and more extraneous frequencies.

Step 4:

 Increase the frequency of the function generator. Note that some of the extraneous frequencies move, but not all in the same direction.

Step 5:

 Continue to increase the frequency towards 5 kHz. What happens when the frequency passes 5 kHz?

Step 6:

 Continue increasing the frequency towards 10 kHz. The behavior of the upper (waveform) display should be similar to what you saw in Part 1.

Step 7:

 Return the frequency to 500 Hz. Slowly increase the frequency and watch the harmonics (of the impure sine wave) as they approach the right hand edge of the display. (Note that this is 5000 Hz, which is half the sampling frequency.) What happens to each harmonic as it reaches this frequency?

Step 8:

 Try to follow one of the wayward harmonics after it is reflected off the right hand edge of the display. What happens to it when it reaches the left hand edge?

Step 9:

 Switch the function generator to triangular wave. Note the waveform and spectrum display. Vary the frequency slightly and watch what happens to the spectrum. Do the same thing with a square wave.

Question 3:

 Explain the spectrum displays you have seen for the "sine", triangle, and square waves. Sketch the spectrum you would expect from a theoretical viewpoint for several "interesting" cases.

Step 10:

 Switch the function generator back to sine wave and set the frequency to 500 Hz.

Part 3: Amplitude Quantization

Once the input signal has been sampled, it must be represented as a number in the computer. Since there are a limited number of bits available to encode the number (12 in this case), there are only a limited number of values that can be exactly represented. Values in between two successive encodings must be rounded or truncated to one or the other. This process of forcing the continuous input range into a discrete set of values is called quantization.


Step 1:

 On the right hand side of the spectrum analyzer is a group of controls which control the quantization of the sampled signal. The Max level control sets the maximum allowed absolute value of the signal. The No of levels control sets the number of different levels that may be represented within the allowed range (i.e. between plus and minus Max level).

Step 2:

 With the quantization button OFF, set the Number of levels control to 100. Turn quantization on by pressing the button. Note the change in the spectrum. Is there any visible change in the waveform display?

Step 3:

 Reduce the number of levels to 10. The effect on the waveform should now be visible. What happens to the spectrum?

Step 4:

 Try several different values for the number of quantization levels. What is the relation between the number of levels and spectrum of the resulting signal?

Step 5:

 Stop and exit the spectrum analyzer program.