SP2012 Signals Spectra Practical

8/2/2019 SP2012 Signals Spectra Practical

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ELEC ENG 3033 Signal Processing / ELEC ENG 7079 Principles of Signal Processing

Practical Signals and Spectra

1 Objectives

The objectives of this practical are:

1. To review the principles of frequency domain analysis, in particular the effects of non-linear opera-tions on signals and their spectra.

2. To empirically verify the theoretical description of mixing by constructing circuits and makingmeasurements using standard School laboratory equipment.

3. To examine the spectral measurements from digital oscilloscope to prompt inquiry into digital signalprocessing principles.

2 Introduction and Background

Fourier analysis uses the principle of describing signals by decomposing them into sinusoidal componentsof several frequencies. This approach to describing signals is highly useful for:

analysing the information contained in a signal;

describing and diagnosing distortion; design of modulation systems;

identifying systems, such as in transfer function measurement.

You should have a good degree of familiarity with the mathematics behind Fourier analysis from yourprevious studies. This practical will require you to incorporate the mathematical knowledge into realobservations in the laboratory situation.

2.1 Revision

Recall from your earlier applied mathematics studies (and if applicable, Signals and Systems II) that aperiodic waveform x(t) of period T can be expressed as a sum of cosine and sine waves,

x(t) =

k=

Xkej2 k

Tt (1)

where Xk can be evaluated using the equation:

Xk =1

T

T0

x(t)ej2k

Ttdt (2)

The magnitude of the component at the index k, |Xk| constitutes a description of the signal in the fre-quency domain, called the spectrum. In a digital signal processor the signal x(t) is represented by samplesx[n] = x(nts), with integer n and ts being the discrete time index and sampling period, respectively. Then

the coefficients are found asXk =

1

N

N1n=0

x[n]ej2k

Tnts (3)

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2.2 Mixers

The mixer, in its simplest form, is a circuit that multiplies two input signals to produce an output, asshown in figure 1.

Figure 1: An ideal mixer circuit.

If we consider x1(t) and x2(t) to be pure (complex) sinusoids of angular frequency 1 and 2, respectively,then the output is

y(t) = x1(t)x2(t) (4)

= ej1tej2t (5)

= ej(1+2)t (6)

In other words, the ideal mixer combines two sinusoidal components to produce a single pure sinusoid atthe sum frequency 1 + 2. For real sinusoids, such as the ones we generate in the laboratory, a furtherdifference frequency term 1 2 arises in the mixer output.

2.3 Non-linear Circuits

In general, a circuit has a characteristic that is non-linear in terms of its input, yNL(t) = f(x(t)), where

f() is an arbitrary mathematical function. When analysing such systems, it is common practice toexpand the characteristic in a Taylor series:

yNL(t) = f(x(t)) = k0 + k1x(t) + k2x(t)2 + k3x(t)

2 + (7)

where kj are constant coefficients in the expansion. For mixing circuits, the key is to exploit the squareterm in the equation above; to see how this works, consider the input to be a sum of two monotonic (i.e.single frequency) complex sinusoid:

y(t) = k2x(t)2 (8)

= k2(ej1t + ej2t)2 (9)

= k2(ej21t + ej22t + 2ej(1+2)t) (10)

Note the first two terms of this output are double the original frequencies, while the last term containsthe sum frequency. If we let 1 and 2 to be very different, then the three frequencies 21, 22 and1 + 2 are well separated in the spectrum and the desired mixing output can be extracted by means ofa suitable bandpass filter.

This concept can be extended to composite signals, i.e. those which comprise of more than one sinusoidalcomponent, is applied to a non-linear device, new components are produced. Their frequencies arerelated to the frequencies of the input components by simple arithmetic. For example, if the inputcontains components at angular frequencies 1 and 2, then the output may contain components atfrequencies such as 1 + 2, 1 2, 21 + 2, 51 32, . . . etc. Such components are denoted by theterm intermodulation distortion and their derivation is not difficult (follow a similar logic to above,and consider the higher order terms).

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3 Spectra of Simple Signals

3.1 Sine Waves

To get yourself familiar with the spectrum measurement feature on your oscilloscope, start with thesimple example of a pure sine wave obtained directly from a signal generator.

Action 1. Examine the spectrum of a sinewave of about 1000 Hz with the oscilloscope. Use the signalgenerator to include an offset equal to the peak value of the sinewave, and observe the result. Tryto become familiar with the controls of the oscilloscope until you can identify all components of thesignal. Take a print out of the spectrum and attach this to your report, and make sure you provide cleardescription of the figure (e.g. appropriate captioning).

3.2 Non-sinusoidal Signals

Now look at the spectra of some non-sinusoidal periodic signals.

Action 2. Examine the spectrum of a square wave and of a triangular wave, again of approximately1000 Hz. Compare the observed spectra with the theoretical spectra for square and triangular waves;find an appropriate reference for the formula you are not required to derive the theoretical spectra.Then insert the plots for these spectra into your report.

Question 1. Examine the square wave spectrum carefully for the presence of any components at evenharmonics. To what could you attribute such components?

4 Building a Mixer to Perform Frequency Shifting

The objective of this part is to examine how frequency shifting can be performed by a simple mixercircuit. The inputs to this mixer are

1. signal generator output with variable frequency

2. a periodic signal generated from a timer circuit

4.1 555 Timer Circuit

The 555 timer is a very versatile integrated circuit that can be used to generate a number of useful signals.In this practical, we will use this IC to generate a periodic signal with variable duty cycle. This requiresthe chip to be configured to operate in the astable mode. Refer to the data sheet to see how the 555 canbe configured to produce a periodic signal with period (i.e. fundamental frequency), with a duty cycle(ratio of the duration of the ON cycle to the overall period) of your own choosing. This is the simplerpart of this practical and you are advised to complete this first.

Action 3. Design a 555 timer circuit to produce a periodic signal with a period of 20 s, with a dutycycle of your own choosing. Construct your design on the SK-10 breadboard, using a 10V DC supply topower the circuit. Test the output from the timer circuit, and attach appropriate plot(s) to your report.Measure the duty cycle obtained from your circuit and verify that it is close to your desired choice. Thencompute the theoretical Fourier series coefficients of the signal and compare the results with the observed

spectrum from the digital oscilloscope.

Once you have completed, tested and documented your timer, you are ready to proceed to the mixer.

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4.2 A FET Mixer

There are many mixer designs in common use. Here, we choose to use a JFET mixer mainly becauseits operation is based on a FET amplifier, and is fairly simple to understand assuming you have donesome Level II electronics. In essence, a FET mixer is a transistor amplifier which has a time-varying biascurrent, which results in a corresponding time-varying gain of the amplifier. If we drive the bias usingone signal x1(t) and apply another signal x2(t) to the input of the amplifier, then the circuits outputwould be proportional to their product, since

y(t) = A(t)x2(t) x1(t)x2(t) (11)

where A(t) is the time-varying gain of the amplifier, which is proportional to the devices transconduc-tance.

To design a basic JFET mixer, consider the circuit in figure 2.

Figure 2: An example JFET mixer circuit.

Assume the active device to be biased in the saturation mode, i.e. the DC bias voltages satisfy VGS > Vt

and VDS > VGS Vt, where Vt is the threshold voltage of the device. In this case, the drain current iDsatisfiesiD = K(vGS Vt)

2 (12)

Since vGS = VGS + (x1(t) x2(t)) (why?), then the achieved drain current contains a component thatis proportional to x1(t)x2(t), thus effecting mixing action. Figure 2 shows the drain to be connectedto a parallel tuned LC circuit which is an optional a bandpass element. When you use such a tunedcircuit, it is important to choose the components L and C such that the output signal only contains thedesired mixing product (e.g. 1 2) while rejecting the unwanted components (e.g. 21). For thispractical, you are required to use a 2N5485 JFET for the construction of the mixer. This is usually anavailable part in the School store, and is sufficient for our purposes. The data sheet reveals that thereare significant variations in the circuit parameters between batches of this component, and so you willfirst need to perform an experiment to characterise your particular device.

Action 4. Connect the 2N5485 in a circuit that enables you to manually vary the Gate-Source voltagewith the DC power supply and to measure the resultant drain current. Plot these measurements and usethem to determine the value of the constants K and Vt in (12) above.

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Once you have obtained the electrical parameters of your JFET device, you are ready to design therequired DC bias network. Basically, this part is very similar to the design of a common source JFETamplifier. That is, you will need to ensure small signal operation, and that the device remains in saturationat all times. Note the blocking capacitors CBL in figure 2 are intended to isolate different DC levels in

various parts of your overall circuit.

Action 5. Perform the design of a JFET bias network. Construct your circuit, including the blockingcapacitors, and test the circuit using two signal generators (you will need to cooperate with your neigh-bouring group in the laboratory). Then connect one of the signals to the 555 timer-generated clock signal.Try applying the timer output to both the Gate or Source of the JFET and note down any observeddifferences. Attach appropriate plot(s) from your circuit to your report.

You are given significant freedom to design the JFET mixer. Normally, JFET mixers are designed witha given mixing gain in mind, but this is not vital for our purposes. The important goal is to achievea circuit that can demonstrate mixing action. Note that the signal levels at the inputs of the mixerare crucial to the success of failure of the circuit. You will need to pay attention to the amplitude of

both the signal generator signal (relatively easy to adjust) and the 555 timer output and may requiresubstantial experimentation. Hint: consider using a potential divider network with low-pass filter for thetimer output signal.

4.3 Demonstrating Frequency Shifting

Now your frequency shifting system is complete, you are ready to examine several scenarios for yoursignal generator output:

1. a pure sine wave

2. a rectangulare wave

3. a triangular wave

Action 6. For all three cases, mix the signal generator output with your 555 timer circuit output, andplot the output spectrum. You will need to choose a suitable frequency for the signal generator outputs;as a guide, frequencies in the kHz range is appropriate. Attach these plots to your report and commenton what you observe, in particular noting any deviations from theoretical expectations.

Question 2. When measuring the output spectrum, zoom in as far as you can to show the individualfrequency components, which should appear as spikes according to theory. Is this what you observe? Canyou come up with a plausible explanation of what you observe? (The real answer to this lies later in thecourse.)

5 Report

To complete this practical, you will need to write a detailed report on the explicit action items. Youwill also need to provide complete yet succinct descriptions of your design, reasoning, observations andconclusions. Your report will not exceed 6 pages in length, excluding appendices, and submission willbe a single PDF file using the electronic submission mechanism on MyUni. More guidelines and tipsfor writing this report will be available on MyUni, and the report will be marked using an assessmentmatrix, to be distributed at least one week before the deadline.

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6 Equipment

The following items are required:

1 Variable DC power supply

1 Digital oscilloscope with FFT capability

1 Signal generator (with square and triangular wave output)

1 SK10 breadboard

1 555 Timer chip

1 2N5485 JFET

Miscellaneous resistors, capacitors and inductors

7 References

Data sheets for the electronic components required for this experiment are found in the Course Material Practical section on MyUni. You will need to carefully read the available information before yourpractical session before doing your designs and during construction of your circuit. Make sure you testeach module of your circuit and verify its correct operation before you jump onto the next section. If youare stuck, then ask the demontrator and/or participate on the MyUni discussion boards.

BWN, Feb 2011. Updated Feb 2012.

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SP2012 Signals Spectra Practical