Signal Recovery TechniquesPHYS 210In the last lab you experimented with generating sounds using an audio card and recording itwith a microphone. You measured the response of the audio system as you stepped through aseries of frequencies. You might have noticed that the resulting data were fairly noisy and easilyaffected by the ambient noise. In this lab we will learn several more sophisticated measurementtechniques that will allow you to separate the signal from the noise. These techniques are used ina wide variety of physics experiments.The general idea is to use as much prior information about the signal as possible. In ourexperiments with the sound system we generated a beep of a known frequency and measured theamplitude of the response. In the first experiment we simply calculated the root mean square ofthe microphone data, which did not take into account the fact that the signal had a single specificfrequency. The r.m.s. was equally affected by the noise in the data at all frequencies. We wantnow to use the prior information about the frequency of the signal to extract it from the noise.1. FiltersThe simplest way to select the signal at a specific frequency is to use a frequency filter.Frequency filters for electrical signals can be easily constructed using simple electroniccomponents, like resistors and capacitors. They can also be implemented in the software at thedata analysis stage.Filters can be divided into low-pass, high-pass, band-pass and band-stop types. As the namesimply, low pass filters allow all frequencies below a certain cut-off frequency to pass through,high-pass filters pass all frequencies above a cut-off frequency. Band-pass filters pass signals ina certain frequency band and band-stop filters reject signals in a particular band, allowing allothers to pass.Filters are never perfect. That is, they can not pass signals in one frequency region undistortedand completely reject signals at other frequencies. The transition from the pass to the rejectregion is usually gradual. Filters can also introduce other undesirable effects, such as a delay anddistortion of the signals.Filters are useful for rejecting noise that is far away in frequency from the signal of interest.Filters are easy to construct and they work in real time, giving a filtered signal without the needfor numerical analysis.2. Fourier TransformsFourier transform is a mathematical operation that separates signals into different frequencycomponents. If S(t) is the signal as a function of time, then the Fourier transform is given by∫−= dtetSFtiωω)()( . The Fourier transform can also be performed on a discrete set of datapoints by replacing the integral with a sum. If the number of data points is equal to an integerpower of 2, the transform is numerically efficient and is called a Fast Fourier Transform (FFT).Since the Fourier transform is a complex number, one often looks at the Fourier TransformPower given by F(ω)F*(ω), which is proportional to the power of an electrical signal S as afunction of frequency.Fourier transforms are convenient for looking at multiple frequency components contained in asignal. They also allow one to measure the strength of the signal at a particular frequency and thenoise in the signal as a function of frequency. One can also implement a filter by multiplying theFourier transform by a certain function of frequency and then performing an inverse Fouriertransforms. Fourier filters have better properties than real-time filters, but they require numericalanalysis and cannot be implemented in real time.The conversion of analog signals to digital form is usually performed by integrated circuitscalled A/D converters. The A/D conversion is characterized by the sampling rate and the digitalresolution in number of bits. If the signal is sampled at a rate equal to R samples/sec, one canshow that the digital form will accurately represent signals with frequencies below fc = R/2.Signals with a frequency f above fc will appear at a frequency 2fc−f. This phenomenon is calledaliasing. To avoid this problem, A/D conversion is usually preceded by an analog low-pass filterwith a cut-off frequency below fc. The digital resolution determines the minimum change in thesignal that can be recorded. For example, a 12-bit A/D converter has a resolution of 1 part in4096. Appearance of discrete steps in the recorded signal is an indication that the digitalresolution is not sufficient.3. Lock-in AmplifierLock-in amplifier is a device that “locks-in” on a signal of a particular frequency and amplifiesit, rejecting all other frequencies. It was invented by Bob Dicke, a professor at Princeton, in1946. To illustrate its operation, lets assume that the signal contains two frequencies, S(t) =A1sin(ω1t+φ) + A2sin(ω2t). In the lock-in amplifier the signal is multiplied by a referencefunction, R(t) = sin(ωrt) and the result is filtered using a low-pass filter. With simpletrigonometric identities one can show that2/]))cos[(])(cos[(2/]))cos[(])(cos[()()()(222111φωωφωωφωωφωω++−+−+++−+−==ttAttAtRtStXrrrrIf the reference frequency ωr is close to ω1, then ω1−ωr is a low frequency that will pass throughthe low-pass filter, while the rest of the terms will be rejected by the filter. In this way the lock-incan select the frequencies close to the reference frequency. If ω1 = ωr and φ = 0, then the lock-insimply measures the amplitude of the signal at ω1. If the two frequencies are slightly different, itmeasures the beats between them.One can also determine the phase of the signal relative to the reference. If we multiply the signalby a reference function R1(t) = cos(ωrt) then we obtain Y(t) = A1sin[(ω1−ωr)t+φ]/2 + (highfrequency terms). If ω1 = ωr, then Y/X = tan(φ). Sometimes it’s also convenient to lookat22YXR += , which does not depend on the phase of the signal.The cut-off frequency of the low-pass filter determines the selectivity of the amplifier. It’s morecommon to refer to the time constant of the lock-in amplifier τ = 1/(2πfc). A long time constant(a small cut-off frequency) is better for rejecting the noise but slows down the response of thelock-in to changes in the signal. A small time constant (high cut-off frequency) increases thespeed of the response but also makes the signal noisier.Lock-in amplifiers can be constructed using analog components, but it’s a fairly complicatedpiece of equipment. However, it is easy to implement a