Lab #7 Questions

One last bit of information about lab #7: These radiation diagrams are calibrated to show the gain of the antenna (the ratio of power density along that vector to the isotropic power density). The directivity is the maximum gain on any vector in the radiation diagram.

Because it is difficult to find the exact spot of maximum gain, you should use the point halfway between the two -3 dB points (in the center of the beamwidth). In the monopole case, you can average the two beamwidths for the two nearly symmetrical sides of the diagram (the error of gain and beamwidth should be at least half the difference between the two sides).

Some questions posed about lab #7:

1) Is there error analysis for this lab? We don't have error's because all of our measurements were discrete.

In your error analysis, you should discuss the relationship of your uncertainty in the antenna directivity measurements, the uncertainty of your beamwidth measurements, and the pointing uncertainty of a real-time communication link. Generally, you cannot align antennae absolutely precisely, but only within the 3dB beamwidth -- how do you account for this in your power budget? How does any allowance you make for beam misdirection compare to the directivity and beamwidth uncertainty?

2) Is there a part A for the lab? When we went to the antenna lab we didn't do any measurements.

Part A uses the radiation diagrams which were provided at the back of the lab report for a horn antenna, a monopole antenna, and a flat panel antenna. You need to calculate the directivity and beamwidth of the first two antennae, and discuss the radiation diagram of the flat panel antenna as it compares to a monopole antenna in a usage such as a handheld mobile unit (a cell phone or wireless Palm, for example).

3) For part E,when we were watching the effective amplitude spectrum of the signals comming for cellular phones we can't determine the number of users on a channel because we don't know aphla and beta and the width of a channel.

You should know the approximate relationship of the width of each peak on the spectrum to the width of each valley (approximately one-third, though due to the resolution bandwidth, the peaks looked somewhat wider). The formula for the number of users in a CDMA channel does not apply -- you should just look at the pulse duration of the IS-136 TDMA phone (about 10 ms) and compare it with the frame period (about 30 ms). A GSM phone would have a frame period about 12 times the pulse duration.

Goal of Lab 4

The goal of this lab was to set up two digital communication systems which used amplitude modulation and frequency modulation, and test whether they worked (if you got out what you got in).

In your abstract, you should clearly state whether the two communication systems worked, what the bit rate you tested was for each system, and how much bias distortion each system introduced. You should also estimate (based on your data) what the bandwidth is for each system WITH ERROR ALLOWANCE.

In the error analysis, for each of the four output waveforms state whether it (within error allowance) does or does not correspond to the input signal (is it a rectangular wave with the same duty cycle). Estimate a range for the bias distortion of the signals in parts C and F; what is the maximum bias distortion that might have occurred? Also use the rise time error to establish the minimum possible bandwidth for all three devices.

For the error range of the highest line on the spectrum analyzer, please use the RMS voltage of the "sampling accuracy" (in the lower right corner of your spectrum analyzer printout). Although it is NOT shown on the error analysis sheet, you should also use the error of the REGENERATOR AMPLITUDE to evaluate your results in parts C and F.

In the discussion, discuss every part of the lab and answer all questions in part G. You may also want to discuss the relationship of the demodulated signal's Absolute Bandwidth (measured with the Spectrum Analyzer in parts B and F) to the bandwidth of each demodulator.

Remember that the above is just a checklist for information you should include in the lab report. It is not a template for your report, so do not simply copy it and insert it into your lab.

And also remember that, because I am providing more information, I expect better reports. So if you are having difficulty understanding this lab, please e-mail me or come by my office hours on Monday.

I have given some tips for writing Lab 3 in office hours today that I want everyone to have available, so I will send this out and post it on the web site. It specifically refers to information for the Abstract and the Discussion.

I have designed this lab to allow for three sets of comparisons among the signals -- the bandwidth of AM signals, the power of the three DSBTC AM signals, and the bandwidth of the DSBSC AM signal vs. the square-wave FM signal (two methods of constant-amplitude modulation). Your abstract should present the BW of the four AM signals. If you want, you can >very briefly< compare them to theory -- in one sentence, state a reason for divergence from theory.

You should then express the power of the three DSBTC AM signals as a fraction of the CARRIER FREQUENCY power of that signal; in other words, the amplitude of the 25 kHz effective amplitude spectrum line squared. Compare these values to theory, but remember that error allowance for a power measurement is TWICE the error allowance for a voltage measurement (since power is proportional to voltage squared). If the error is very high, also consider the error of the modulation index.

Next you should compare the BW of the DSBSC AM signal (part C) to the square-wave baseband FM signal. These signals have the same power. Which makes more effective use of bandwidth?

In the discussion, you should also compare the bandwidth of the FM signal in part E to the theoretical predictions:

(1) of Carson's rule with the square wave absolute bandwidth

(2) of Carson's rule with the information bandwidth of a 2000 samples per second digital signal (this bandwidth is 1000 Hz)

(3) the predictions in Lecture 3, section 12 for an MSK signal.

Note that Carson's rule is most effective for analog signals (as in Part D of this lab). For FSK/MSK systems in general, we will filter the baseband digital signal to reduce its bandwidth before modulation, so that the required FM signal bandwidth is closer to Carson's rule in approximation (2) above than Carson's Rule in approximation (1).

1. How do I treat a pi/2 phase shift? Can I replace all the cosines in the terms with sines and so the amplitudes remain the same and the phase spectrum is zero across the board?

In general, stay with all cosines in your Fourier series, since it makes analysis easier. To set the phase spectrum at zero, make the time-domain function even around the t=0 point. In other words, for a square wave, have half the "mark" (positive voltage) on the left side of the t=0 point (the y-axis) and the other half of the "mark" on the right side of the t=0 point. For a square wave, this will result in all signals having a phase of 0 (with cosine components of the Fourier series).

2. What does a phase shift greater than pi mean?

This is just a way of expressing all amplitudes as POSITIVE values in the amplitude spectrum. For a square waveform, this means that the third, seventh, eleventh, et cetera harmonics have POSITIVE amplitude but a 180 degree phase shift.

An harmonic with a negative amplitude and a 45 degree phase shift can also be expressed as an harmonic with POSITIVE amplitude but a 225 degree phase shift (phase 180 degrees opposed to the original phase).

The reason to think of phase shifts in the range 0 to (2 pi) is that we normally express the amplitude spectrum as a POWER spectrum, so all values are positive (RMS values). If we have the POWER spectrum and the [0 to (2 pi)] phase spectrum, we can reconstruct the time-domain waveform.

3. What is a rectified sine wave? What does it look like?

A "half-wave rectified" sine signal is a sinusoidal signal where any negative voltages are replaced by a 0 voltage. See page 26 of the green book, the third diagram. This shows a "half-wave rectified" cosine signal -- a "half wave rectified" sine signal is just this same signal shifted by pi/2 radians. Since we aren't measuring the signal in the time domain (we're measuring only amplitudes in the frequency domain), this phase shift does not matter.

A "fully rectified" cosine signal is a sinusoidal signal where any negative voltages are inverted. Where the time-domain trace is flat in the figure on page 26, there would be a repeat of the pattern from the first half-wave. We are not using a fully rectified signal in this lab.

4. When you refer to amplitude are you refering to lowest values to highest or from zero to highest?

In this class, "amplitude" always refers to the midpoint-to-peak voltage difference. I will very often say "peak-to-peak" amplitude -- this is the bottom-to-top measurement, which for most waves should be twice the amplitude. On the digital oscilloscope, the "Amplitude" measurement is actually bottom-to-top, though -- it's the same as "Peak-to-Peak", but with a slightly different definition for the "Peak".

Of course, there are exceptions. The amplitude of a "half-wave rectified" sinusoidal signal (page 26) is measured baseline-to-peak -- this is because it has the same amplitude as the sinusoidal wave that created it.

For a digital signal, the amplitude is often expressed as the space-to-mark (zero-to-one) distance. The square and rectangular wave formulae on page 26 use the space-to-mark amplitude. Remember that if your have the space-to-mark voltage difference, you must divide by 2 to get the amplitude for these formulae. And if the square wave has a mark at voltage = zero, you must add a DC offset to the Fourier series.

5. How do we calculate the RMS or effective amplitude?

The Fourier Series for these signals will show the actual amplitude. Use this amplitude in the amplitude spectra. The RMS voltages are just the absolute values of the amplitudes divided by the square root of 2. Use the RMS voltages in diagramming the EFFECTIVE amplitude spectrum. Also use RMS voltages in the theoretical predictions in the results section.

6. In the rectangular waves, do we have to worry about the phase shift?

No, we don't have to worry about the phase shift for any rectangular waves. (The easiest way to ignore a phase shift is to center the mark of any rectangular wave around the time=zero point, and then use only cosines to describe the wave.)

Prelab 1, First Question:

A harmonic is a spectrum line whose frequency is an integer multiple of the signal's fundamental frequency (1000 Hz for most of this experiment). A periodic signal will have spectrum lines only at these harmonics; depending on the wave shape, some harmonics may be "missing", with a predicted amplitude of zero.

If you have a square wave with f=1000 Hz and amplitude of 4V, what will its frequency domain representation look like? What will the RMS voltage values of the harmonics be?

For a square wave (50% duty cycle) with frequency f=1000 Hz and amplitude 4V, the equation on page 31 of the green book will give you the amplitude spectrum (not effective amplitude), where w1, 3w1, 5w1, ... are the angular frequencies of its component waves (harmonics). To get you started, the w1 value represents 1000 Hz * 2(PI). The next harmonic will be three times this value. The amplitude of the w1 component (the first harmonic) will be 4A/(PI). The other amplitudes follow from this.

Prelab 1, Second Question:

The lab does not refer to angular frequencies. The frequencies listed in the lab (1000 Hz, et cetera) are 1/period. The angular frequency for a 1000 Hz wave would be 6283.2 1/s. Any time that "Hertz" is used, it refers to the frequency, not the angular frequency.

An "amplitude spectrum" contains spectrum lines whose height is the actual amplitude of that harmonic. An "effective amplitude spectrum" contains spectrum lines whose height is the the RMS amplitude of that harmonic (the absolute value of the actual amplitude divided by the square root of 2).

Prelab 1, Raw Data:

You should put in the raw data section the predicted frequency domain representation (using effective amplitude, or RMS voltages) of the given signal.

If a wave has a DC component (for instance, a rectangular wave with duty cycle <> 50%), put this down as the "0th harmonic".

For each wave, provide enough harmonics to get past the full bandwidth of the signal. Include the harmonics with zero amplitude -- you may see a small value for these harmonics when you perform the experiment.

To calculate the bandwidth, sum up the power of each harmonic of the signal (the RMS voltage squared), starting with the first harmonic. Do not use the DC component in the bandwidth calculation. Add the power of each new harmonic to the total, then divide the harmonic's power by the new total. If it's less than 50%, you're outside the EFFECTIVE bandwidth. If it's less than 2%, you're outside the FULL (or absolute) bandwidth. So keep adding harmonics until you reach a harmonic whose power is less than 2% of all the harmonics summed.

Measure the bandwidth from zero frequency to the highest frequency line measured above. These signals are "baseband" signals; they directly convey information with the signal voltage. In labs 3, 4 and 5, we will see how the bandwidth of the baseband signal determines the bandwidth of an amplitude modulated signal and a frequency modulated signal.

Lab 2 Error Analysis:

1. Why do the Bode Plots (both amplitude and frequency) differ between the ideal case and the experimental result? The experimental amplitude Bode Plot keeps going downward at higher frequencies.

Look at the behavior of an RC low-pass filter in the lecture notes. Note that there is both attenuation and phase-shifting. If you were to add an RC filter to your theoretical Bode plot, would it start to look like your experimental result?

2. In parts B and C of the experiment, we got both positive and negative values for time delay, and these values are far from the expected theoretical value, 0.76 x 10-6 s. Was the experiment performed incorrectly?

Remember that the time delay does not measure the number of COMPLETE CYCLES that the wave has been delayed. If you add the additional time for the number of complete cycles (n x cycle time) you should obtain a fairly uniform value for time delay over all frequencies.

3. What calulation should be included in the error anaylsis in addition to the bandwidth error? Do we need to find the maximum relative error for time delay or the attenuation?

You should estimate the error of your phase shift measurements (the digital oscilloscope error plus the error of your length measurement plus the frequency error of the function generator). Compare this to the deviation of your Bode plot from the expected. This should be more qualitative than quantitative -- you want to verify whether the phase error corresponds to the expected value within acceptable error. If it doesn't, seek an explanation why (see #1 above).