STATIC AND DYNAMIC CHARACTERISTICS OF SIGNALS INPUT sensor transducer Signal A MEASUREMENT SYSTEM Signal B OUTPUT Signal C Signal just associated with transmission of information Important Questions What is the dynamic range of the measurement system What is the frequency response of the measurement system What are the characteristic of the signal from your sensor transducer Are you looking for a static or dynamic response Classification of some basic types of signals based on acquisition representation 1200 Price dollars 1000 800 600 400 200 0 0 1000 2000 Days Ago Reminder How many discrete quantization levels are there in a signal digitized by an 8 bit analog to digital converter 256 How do we analyze signals Magnitude and Phase single frequency Frequency Content Fourier analysis Apply signal processing techniques filtering time averaging Very simple signal analysis may be suitable for some tasks What is the mean value of a signal For discrete data this is simply t2 y y t dt t1 t2 N Area under curve dt t1 Time window y y n 1 N n The average value is a function of the time window select the time window based on measurement plan If you are interested in the AC component of a signal the simple operation of subtracting out the DC component can be useful for analysis Another useful quantity for characterizing the dynamic portion of a signal is the Root Mean Square Value RMS t2 yRMS 1 2 y dt t2 t1 t1 yRMS 1 N N 2 y n n 1 What does this look like Let s look at a mass spring system d2y m 2 ky 0 dt Where does this come from The general solution is y A cos t B sin t How do you find these y C cos t harmonic oscillation k resonance m Using some trig we can C Also write the solution as A2 B 2 B A tan A What s going on here We are splitting light into it s individual frequency components f Low spatial frequency filter Lower spatial frequency filter Block low frequencies Block more low frequencies What is a periodic function f x T f x period If f x and g x are periodic with the period T then a linear combination of the two is also periodic h x Af x Bg x We can represent any function of period 2p in terms of simple functions this can help us in evaluating a signal solving complex equations evaluating a system response Use sin x cos x sin 2 x cos 2 x sin nx cos nx Note the primitive period of these are different but all have a period of 2p To represent a given function we form a trigonometric series a0 a1 cos x b1 sin x a2 cos 2 x b2 sin 2 x an cos nx bn sin nx a and b are the coefficients of the series Great so we can write this as f x a0 an cos nx bn sin nx n 1 But we need to find these coefficients Fortunately we can follow an established strategy 1 For a0 we integrate both sides from p to p p p p p f x dx p a dx p a 0 p p n 1 n p p f x dx p a dx p a 0 n 1 p p n cos nx dx 0 0 p b n sin nx dx p p f x dx p a dx 2p a cos nx bn sin nx dx or 1 a0 2p p p f x dx We have the first coefficient Ok let s continue with an f x a0 avn cos nx bn sin nx n 1 Let s try multiplying both sides by cos mx where m is a positive integer and integrating both sides from p to p p p p f x cos mx dx p a 0 0 p cos mx dx an cos nx cos mx dx n 1 p p p b n cos mx sin nx dx Do any of these terms go to zero Let s examine the other terms p a an cos nx cos mx dx n 2 p p 0 an cos n m x dx p 2 p p cos n m x dx Only non zero when n m OK so let s look at the last term p p p f x cos mx dx p a 0 p cos mx dx an cos nx cos mx dx n 1 p p bn bn sin nx cos mx dx 2 p 0 bn p sin n m x dx 2 p p p b n cos mx sin nx dx 0 sin n m x dx p p So it looks like we only have nonzero terms for n m p p a f x cos mx dx n 2 an 1 p p cos n m x dx p 2p an p an 2 p p f x cos nx dx We have all an coefficients We can do a similar trick by multiplying by sin mx to find bn 1 p p p f x sin nx dx We have derived the Euler formulas f x a0 an cos nx bn sin nx a0 an bn 1 2p 1 p 1 p p p p f x dx p f x cos nx dx n 1 Note periodic function 2p can integrate over any 2p period p p f x sin nx dx For a periodic function f x the Euler formulas allow us to represent the function as a Fourier Series the coefficients that we found are called the Fourier coefficients k p x 0 f x k 0 x p f x 2p f x Let s start with a0 1 a0 2p p p f x dx k p p k 0 p 1 a0 K dx k dx 0 2p p 0 Note that this coefficient just gives the DC value of the signal In this case the DC value is zero x k p x 0 f x k 0 x p How About an Plug in Simplify f x 2p f x an 1 p p p f x cos nx dt 0 p 1 an k cos nx dt k cos nx dt p p 0 an 1 k k 0 p sin nx sin nx p 0 0 p n n Wow All of the an terms are zero How About bn bn 1 p p p f x sin nx dx Plug in Simplify 0 p 1 bn k sin nx dt k sin nx dt p p 0 bn 1 k k 0 p cos nx cos nx p 0 p n n k 1 cos np cos np 1 np 1 n even 2k cos n p 1 cos np np 1 n odd b1 What are the Fourier Coefficients 4k p b2 0 b3 4k 3p Interesting But what does this mean f x a0 an cos nx bn sin nx n 1 f x bn sin nx n 1 b1 4k p …