EXPERIMENT 4 SIGNAL RECOVERY

Transcription

1 EXPERIMENT 4 SIGNAL RECOVERY References: A. de Sa, Principles of electronic instrumentation P. Horowitz and W. Hill, The art of electronics R. Bracewell, The Fourier transform and its applications E. Brigham, The Fast Fourier Transform 1 Aim You will investigate some of the main techniques that can be used to recover either analogue or digital signals buried in noise. Some of these techniques, such as correlation and transform methods, are also used in some more sophisticated applications such as data processing of noisy images and complex waveform retrieval in audio, video and ultrasonic applications. 2 Introduction In most areas of science, medicine and engineering there is an increasing requirement to measure small or complex signals that are often obscured by noise or other forms of interference. The recovery of a signal of interest that is obscured by noise is becoming the limiting 1

2 process in many applications particularly since modern data acquisition techniques make capture and processing of the final signal of interest quite straightforward. Thermal noise arises from the random motion of charges in a conductor at a particular temperature and is a consequence of the second law of thermodynamics. Nyquist showed that the root mean square (rms) noise voltage V rms in a conductor of resistance R at a temperature T is V rms = (4kT RB) 1/2 (1) where k is Boltzmann s constant and B is the noise equivalent bandwidth. The noise power is then P n = 4kT B which is independent of resistance and frequency. Hence, this noise is constant over the frequency spectrum and is referred to as white noise. Shot noise is due to the random fluctuations in the emission of charge from a thermionic source. This emission obeys Poisson statistics and the rms noise current is I rms = (2qI eq B) 1/2 (2) where q is the charge and I eq is the equivalent saturated current. In addition to these fundamental noise sources, a number of forms of environmental noise also exist including mains interference and transients, cross-talk between signal lines, mechanical and acoustic noise, radio frequency and other electromagnetic interference, power or inductive switching spikes, cosmic radiation and fluctuations due to natural phenomena such as thunderstorms or lightning. Aspects of this experiment also relate to the experiments on Noise and on Fourier analysis. 2.1 Equipment Circuit modules: Four preassembled modules (summing amplifier, autocorrelator, lock-in amplifier and exponential decay generator) are provided together with the appropriate power supplies. In addition to the above, the following components will be available: ultrasonic devices, BNC and BNC/RCA cables, CRO probes and an optical rail with mounts. Noise generator: The Rohde & Schwarz SUF 2 noise generator will be used to obtain white noise of variable amplitude and bandwidth. The output attenuator covers a range of 0 to -100 db and the noise bandwidth is selectable for 110 khz, 6 MHz and 50 MHz upper bandwidth limits. Function generator: The Hameg HM function generator can be used to derive the signal of interest. Digital oscilloscope: The LeCroy 9310 is a sophisticated digital CRO with a 300 MHz bandwidth, sampling rates up to 100 Msamples/s single-shot or 10 Gsamples/s repetitive, various external interfacing options and extensive processing capabilities. This CRO is fitted with a RAM card reader and both fast Fourier transform (FFT) and waveform processing software packages. Computer: A PC will be used to run simulation software and interface with the CRO. Parallel bi-directional communication between the oscilloscope and the computer is achieved using the IEEE-488 bus and includes an IEEE-488 interface card in the PC. The simulation software comprises three applications each run under LabView TM. The program autocorrelation evaluates the autocorrelation function for selectable noisy waveforms. The program Fourier Filtering illustrates the technique of Fourier transformation, filtering the Fourier spectrum of a complex signal and reconstruction of the signal of interest by inverse Fourier 2

3 transformation. Finally, the program lock-in amplifier demonstrates the principles of operation of the lock-in amplifier. Lock-in amplifier: A commercial lock-in amplifier (Stanford SR 510) will be used for the ultrasonics application. The performance of the SR 510 may be evaluated and compared with the simpler circuit module lock-in amplifier. Box-car integrator: This item is a Stanford Research Systems box-car integrator system comprising a SR250 gated integrator/box-car averager, SR245 computer interface, SR200 gate delay scanner and a power supply unit all housed in a NIM rack. The display unit incorporates a voltmeter with analogue, digital and bargraph display modes, and can be used for monitoring various dc voltages during this experiment. 3 Procedure 3.1 Getting to know the equipment In order to investigate the noise reduction characteristics of signal recovery techniques, we require a signal source which comprises both a signal of interest, S i, and a noise signal, S n, whose amplitudes can be independently adjusted to obtain a total signal S = S i S n where the signal-to-noise ratio, SNR = S i /S n, is variable. We can do this by mixing the output signals from the function generator and the noise generator using the summing amplifier, shown in Fig. 1. Adjusting the output attenuators of the noise and function generators then All resistors 1 k 1% IC 1 = LF353, TL072 or equivalent signal noise - V 1 F 8 2 IC 1a F V IC 1b 5 7 Figure 1: Summing amplifier for combining signal and noise. allows easy and continuous setting of the signal, noise and SNR levels. The effectiveness of a noise reduction method can be characterised by the signal-to-noise improvement ratio SNIR = SNR output / SNR input. Learning to use the digital oscilloscope The oscilloscope will be used continuously throughout this experiment and it is recommended that you spend some time becoming familiar with its operation. The blue auto setup button will automatically display the input waveform. The dark grey push buttons on the front panel will select various functions and the appropriate information displayed on the right-hand side of the screen. Settings can then be changed using the white buttons adjacent to these displayed menus. Input coupling, triggering, timebase and attenuator 3

4 settings can then be chosen to obtain the appropriate operation. Mathematical operations, waveform processing and FFT facilities are accessed using math setup and then defining math memories A-D. Voltage, time, statistical and other parameters calculated for the waveform can be obtained using cursors/ measure. Capture of the oscilloscope traces by the computer is done using the acquire function within the Labview TM -based programs and waveforms may be printed from these programs using the print function. Choose a standard noisy input signal comprising a sinewave with a particular amplitude and frequency together with noise of a particular amplitude. A signal frequency 1 khz, SNR 0.05 and noise level of 0 db are recommended. Use this standard signal in the different parts of the experiment to compare the effectiveness of different signal recovery methods. 3.2 Signal averaging The SNR for a noisy repetitive waveform can be improved by repetitive averaging provided that the averaging is triggered on the same point of the waveform of interest for each sweep. Averaging can be achieved using the averaging facility on the CRO. Combine noise (0 db attenuator setting) with either a sine or square wave such that the noise totally obscures the waveform of interest (SNR 0.05). Display both the noisy and averaged signals, and observe the improvement in SNR as you increase the number of sweeps averaged. You can calculate the SNR by finding the rms value of each signal separately then divide the noise rms into the signal rms to obtain the SNR. Observe how the sine wave that is buried in the noise can be made visible with increasing the number of averages and therefore increasing the SNR QUESTION: What is the rms value of a pure sinewave with amplitude A? What about a square wave with amplitude A? Set the signal amplitude to zero. Determine the rms noise level of the averaged signal as a function of the number of sweeps averaged. The rms voltage of the averaged signal, V rms, is dependent on the number of averages, N, by V rms N a. Determine the exponent, a, by a graphical technique. Although the noise reduction you have just carried out had a zero input signal, in practice there usually is a signal of some type. As a result, this leads to an improvement in the SNR because noise is random and in the long term should average to zero. However, the signal is repetitive and should stay as it is provided the oscilloscope is triggered from the same position on signal. Optional advanced question: Can you find an explanation in the reduction in noise rms based on statistics? C1 Tutor checkpoint. Obtain tutor s signature before proceeding. 4

5 3.3 Correlation techniques Most noise reduction techniques inherently involve the use of correlation of the noisy signal with some function or reference. The cross-correlation function for two functions f and g is defined as C fg (τ) = f(t τ)g(t)dt (3) Note that in mathematics this is known as the convolution. If f and g are the same, we obtain the auto-correlation function C f (τ) = f(t τ)f(t)dt (4) Performing a correlation of a noisy signal with itself leads to generation of the autocorrelation function of the signal of interest and the suppression of the uncorrelated random noise. This process should therefore enable an enhancement of the SNR. Open the program autocorrelation. Select, in turn, various functional forms for the input waveform which will then be displayed in the upper panel and observe the autocorrelation function in the lower panel. You may wish to analytically evaluate the integral C f (τ) for one or more functions f(t) and compare with the computer-generated result. Finally, add noise. Draw what you observe and comment about it. 3.4 Fourier filtering A continuous function, f(t), can be separated into its various frequency components using the Fourier transform F (ω) given by and the inverse transform is F (ω) = f(t) = f(t) exp(iωt)dt (5) F (ω) exp( iωt)dω (6) If f(t) is determined only at discrete equally-spaced intervals of t = kδ, k = 0, 1,...N 1 then the discrete Fourier transform is defined as N 1 F (ω n ) = δ f k exp(2πikn/n) (7) k=0 where ω n = 2πn/(Nδ), and n = N/2,...N/2. The Fast Fourier Transform is an efficient algorithm for the evaluation of the above summation and is described in detail in the reference texts. Open the program Fourier Filtering. 5

6 An input signal can be synthesised from sine, square and chirped waveforms of selectable amplitude, frequency, phase and duty cycle using the software generator G and displayed in the top panel. Gaussian or white noise of variable amplitude can be added to this waveform. A switch selects either the signal or the signal noise. The FFT is displayed in the centre panel and the spectrum can be filtered by choosing:- level which rejects frequency components with amplitudes below a given level, frequency which allows switch-selectable high-pass or low-pass filtering, range which allows switch-selectable band-pass or band-reject filtering. The chosen levels or cut-off frequencies are obtained by dragging the cursors to the desired positions. The build function will reconstruct the input and recalculate the FFT. After selecting the filter type, process will apply this filter to the FFT and calculate the inverse transform which will be displayed in the bottom panel. Experiment with various filter types for several input waveforms and SNR settings. After becoming familar with the software, input your standard noisy signal into the oscilloscope and capture this using the acquire button on the screen. Experiment on real data with the different filter types to recover the signal from noise. Print a selection of results for your logbook including each of the filter types and for both synthetic and real data. C2 Tutor checkpoint. Obtain tutor s signature before proceeding. 3.5 Lock-in amplifiers This technique gives you a dc signal that is proportional to the amplitude of the input signal. The lock-in amplifier is so-named as it amplifies a signal and locks in to that part of the signal at the frequency of interest. Since noise has a broad band of frequencies, then a large part of the noise signal will be greatly reduced. The technique can be used with continuous repetitive waveforms and can yield extremely high SNIRs. The technique requires a reference signal that is at the same frequency as the signal of interest and which has a stable phase relationship with the signal of interest. Specifically, the signal and reference frequencies are multiplied, which results in a signal with high and low frequency components. A low pass filter removes the high frequency components. The low frequency components (ideally the dc component) is proportional to the amplitude of the original noise free signal. The cut-off frequency of the low pass filter is determined by the time constant setting on the lock-in amplifier. This time constant is similar to the time constant in circuits that contain a resistance and a capacitance. In technical language this method uses synchronous demodulation (sometimes called phase-sensitive detection or synchronous rectification) to extract that component of the signal that is both at the relevant (reference) frequency and which is phase-coherent with the reference signal. The noise bandwidth f n of the device is effectively determined by the time constant (1/ f n ) of a low-pass output filter. Thus, the dc output voltage is related to the amplitude of the phase-coherent component of the input signal at the reference frequency. Now investigate the function of a lock-in amplifier using the software simulation of the lock-in amplifier. 6

7 Measure the dc output level as a function of the phase of the input sinewave without noise present. Plot the output dc level V as a function of cosφ and show that the maximum sensitivity is obtained when the reference and signal are in-phase, and that V = V 0 cos φ. Add noise and record the dc output level for the full range of input noise levels. Using these results, describe the principles of operation of the lock-in amplifier, the role of the multiplication and averaging steps and the noise rejection characteristics. Now we will see the same principles in operation using the SR510 Stanford Systems commercial lock-in amplifier. Connect the signal generator to both the reference input and input A. Attenuate the signal so that the output does not overload. Observe the cosine dependence of the output as the phase between these inputs is continuously varied. We will now use this system to extract a small signal from a noisy environment using ultrasound transceivers Application of a lock-in amplifier to signal recovery in ultrasonics Ultrasonic systems are widely used in medical imaging and industrial inspection. Noise reduction methods are often required because most materials cause a large attenuation of the ultrasonic intensity. In our case the material will be air. function generator CRO transmitter receiver signal input lock-in amplifier reference output Figure 2: Ultrasound signal detection. Set up the experiment shown in Fig. 2 using a modulation frequency of 40 khz. Adjust this reference frequency to obtain maximum sensitivity and note the frequency. Display both the transmitter and receiver waveforms on the CRO and note the relative phase shift when their separation is changed. 7

8 Using this effect, measure the wavelength and hence the velocity of these ultrasonic waves. Compare your value for the velocity with the theoretical value using the formula below: V = γp ρ = γrt M where T is the absolute temperature, M is the average molecular weight of the air and R is the universal gas constant, γ is the exponent for adiabatic processes in the formula P V =constant and can be taken as approximately 1.4 at room temperatures. (This means we assume translation and rotation are possible ways of storing energy in the diatomic gas molecules but that the vibrational mode is not invoked.) Now plot the amplitude and intensity as a function of separation between the transmitter and receiver. Remember to re-adjust the phase for maximum reading each time you take a measurement. Remember, the ultrasound transmitter looks like a large number of single point transmitters when viewed very close. However, at very large distances it will look like a single point transmitter. These two regions are known as near and far fields, respectively. Comment if you can determine whether there is near and far field behaviour from your plots. Determine the power law dependence on distance of the far field. QUESTION: Is the power law dependence of the far field what you might expect? Comment. C3 Tutor checkpoint. Obtain tutor s signature before proceeding. 3.6 Optional further work - Box-car integrators The box-car averager (Fig. 3) uses an analogue gate or switch to control passage of the buffered input signal to an RC low-pass filter which integrates or averages that part of the input waveform that is acquired when the gate is closed. A trigger pulse defines a time reference point and from this is derived an adjustable gate of width t w and delay t d which determine the analogue switch closure time and duration. Therefore, the box-car will average the input signal during the period t w following the delay t d after the trigger pulse. Furthermore, by choosing a suitable t w and scanning t d from zero to the length of the signal, the waveform can be reconstructed with improved SNR. Box-car integrators are used to measure transient phenomena occurring as repetitive signals typically at a low repetition rate. These are often exponential or other decays following a fast rise in the signal. Such a waveform is generated by the circuit in Fig. 4. The output of this generator can be fed to the mixer together with noise and the resulting noisy exponential decay signal then used as the input signal to the box-car integrator. The negative sync output pulse from the waveform generator can then be used to trigger the box-car. The gate delay can be scanned using the gate scanning module or, more easily, by manually adjusting the gate delay switch and multiplier vernier. The box-car output can be displayed on both the CRO and the display unit. Connect the output of the exponential decay generator and the output of the noise generator to the mixer. 8 (8)

9 Figure 3: The box-car integrator. With the noise generator disabled, demonstrate that the box-car output tracks the input signal waveform as the delay is varied for reasonable settings of the gate width and sample rate. Using a noise generator setting of 0 db, measure the time constant of the decay and compare it with that measured directly at the output of the waveform generator. This is most easily done by taking measurements at say two suitable delay times and using a suitable gate width and number of samples. C4 Tutor checkpoint. Obtain tutor s signature. 9

10 5V 1 F 100 khz TTL trigger input NE555N nf 39 k 10 nf - V 8 5 IC 1a 6 7 IC1 = LF353, TL072 or equivalent 7V 1 F 15 k 1 M IC 1b 1 F 3.9 k 4 1 V - 1 k 1 k 100 k 5V, 5 s sync output pulse to trigger box-car -7V 500 mv, 2.3 khz repetitive exponential decay waveform Figure 4: Exponential signal generator. 10