Chapter 2

Static and Dynamic Characteristics of Signals

Signals

„ Signals classified as 1. Analog – continuous in time and takes on any magnitude in range of operations 2. Discrete Time – measuring a continuous variable at finite time intervals 3. Digital – discretized magnitude at discrete times

„ Quantization assigns a single number to represent a range of magnitudes of a continuous signal

„ This is done by a Analog to Digital A/D converter

1 Signal Wave Form

„ Static – does not vary with time and can be considered over long periods, like the life of a battery. „ Dynamic Signal – time dependent y(t)

Definitions

„ Deterministic Signal –a signal that varies in time in a predictable manner, such as a sine wave, a step function, or a ramp function. „ Steady Periodic –a signal whose variation of the magnitude of the signal repeats at regular intervals in time.

Definitions cont’d

„ Complex Periodic Waveform – contains multiple frequencies and is represented as a superposition of multiple simple periodic waveforms. „ Aperiodic Waveforms – deterministic signals that do not repeat at regular intervals (step function). „ Nondeterministic Signal –a signal with no discernible pattern of repetition.

2 For periodic signal: c = amplitude f = 1/T cycles/sec (hz) T = period ω = 2Πfrad/sec

Signal Analysis

„ Assume y=current t 2 t 2 Y = (())/()ytdt dt ∫∫t1 t1 If power is equal to P=I2R, then energy is dissipated in resistor

from t1 to t2

Signal Analysis

t 2 T 2 „ E ==Pdt [()]It2 Rdt ∫∫t1 T1

„ To find a fixed current Ie that produces some energy dissipation in time t1 to t2, 2 2 2 t t 22 t 2 E ==Pdt IRdtIRtee =−=()[()]21 t It Rdt ∫∫t1 t1 ∫t1

2 t 2 ItteRMS=−(/(1 21 )) [()]It dt ⇒ I ∫ t1

3 „ A time dependent analog signal can be approximated by a discrete set of N numbers over the time period

from t1 to t2 y(t) {y(rδt)} r=1,2,N where δt = sampling time

Mean Value

„ For discrete time signals, approximate the mean value N yNy= (/1 )∑ i i=1 RMS

N 2 yNyrms= (/1 )∑ i i=1

Signal Averaging Period

„ The choice of time period for signal analysis depends on the type of signal „ For simple periodic, use time period equal to the period of the function. That gives a good representation of long term average value and RMS value.

4 Signal Averaging Period

„ Averaging a simple periodic signal over a time period that is not exactly the period of the function can produce misleading results. „ However, as the averaging time period becomes long relative to the signal period, the resulting values will accurately represent the signal.

Signal Averaging Period

„ Complex wave forms and non- deterministic signals do not have an exact waveform time period because they are made up of signal components with different frequencies.

„ Remember: f=1/T hertz

Effect of Time Period Selection

„ In this case select a sampling period that is large with respect to the longest frequency.

5 Effect of DC Offset If AC component is of primary interest, subtract out DC component, then amplify AC to get more pronounced impact from AC.

Signal Amplitude and Frequency

„ Basis: very complex signals, even non- deterministic, can be approximated by a series of sine and cosine functions (fourier analysis) „ Fourier Analysis is a mathematical equivalent of a prism.

Signal Amplitude and Frequency

„ “the representation of complex and non- deterministic waveforms by a simple periodic function allows measurement system response to be reasonably well defined by examining the output resulting from a few specific input waveforms, one of which is the simple periodic.”

6 How do you represent a complex signal by simple function?

„ The fundamental concepts of frequency and amplitude can be understood through the observation and analysis of periodic motions. „ Sines and cosines can be thought of as mathematical functions that describe specific physical behaviors of a system. „ Consider the mechanical vibration of a spring-mass system

Spring-Mass System

„ Governing Equation: M(d2y/dt2)+ky=0

„ General form of solution: y=A cos wt+ B sin wt ; ω= km/ T= time of one cycle, called a period f= 1/T hz w= 2Πfrad/sec

7 Frequency Analysis

„ The solution can be rewritten by use of phase angle: y=c cos (wt-ϕ) or y=c sin (wt+ϕ*) Where caB=+22; ϕ=tan-1(B/A); ϕ*=Π/2 - ϕ

Frequency Analysis

„ Frequency Analysis: most signals from measurement systems are non-deterministic, having complex wave forms and consequentially can be broken into series of sine and cosine functions. „ The representation of a signal by an infinite series of sines and cosines is called a fourier series. „ In theory, all mathematical functions can be represented by a fourier series.

Definitions:

1. Y(t) is a periodic function if there is a positive number T y(t+T)=y(t)

-If both y1(t), and y2(t) have period T then ay1 + by2 have period T

2. A trigonometric series is given by:

Ao + A1 cos t + B1 sin t +A2 cos 2t + B2 sin 2t +…+ 1 where An and Bn are coefficients

8 „ Example of harmonics of a string plucked at center:

Fourier Series and Coefficients

„ A periodic function∞ y(t) with period T=2Π given as yt()=+ Aon∑ ( A cos nt + B n sin nt ) n=1

„ With y(t) known, determine Ao, An, Bn Integrate y(t) from -Π to Π πππ ∞ π ∫∫∫ydtAodt((t)cossin)= ++∑ An ntdt Bn ∫ntdt −π −π n=1 −−π π Since π π cosntdt = 0 and sinntdt = 0 ∫ − π ∫ − π

Euler’s formula: π Aytdto = 12/(π ) ( ) ∫ − π π A = 1/π y(tntdt )cos Follows thatn ∫ −π and π B = 1/()sinπ yntdtt n ∫ −π

„ However,most functions do not have a period of 2Π that will be encountered.

9 Fourier Coefficients for Functions Having Arbitrary Periods

Coefficients representing a function of frequency ω are given by: T / 2 ATytdto = 1/() ∫ − T / 2 T / 2 ATytnwtdtn = (/)2 ()cos ∫ − T / 2 T / 2 BTytnwtdtn = (/)2 ()sin ∫ − T / 2 Where n = 1,2,3,…, and T = 2π/ω

Fourier Coefficients for Functions Having Arbitrary Periods

„ The trigonometric series that results from these coefficients is a Fourier series and may be written as: ∞ y(t) = A0 + ∑(An cosnωt + Bn sinnωt) n=1

Fourier Coefficients for functions having Arbitrary Periods

∞

The series yt()=+ Aon∑ (An cosnwt + B sin nwt ) can n=1 be rewritten using phase angles as

∞ yt( )=+ AonC cos(nwt −φ ) 22 ∑ n Where CABn = nn n=1 tanϕ nnn= B / A ∞ tanϕ * nnn= AB / + * yt()=+ Aon∑Cn sin(nwt φ ) n=1

10 Even and Odd Functions

„ A function g(t) is even if it is symmetric about the origin for all t, „ g(-t) = g(t) „ A function h(t) is odd if, for all t, „ h(-t) = -h(t)

Fourier Coefficients for functions having Arbitrary Periods

If the signal is even, Bn=O and the series becomes: ∞ ∞ yt()= ∑ An cosnωt yt()= ∑ An cos(( n2π t )/ T ) n=1 n=1

If the signal is odd, An=0 and series becomes: ∞ ∞ yt()= ∑ Bn sinnωt yt()= ∑ Bn sin(( n2π t )/ T ) n=1 n=1

Example 2.5:

„ Original: E(t)=120 sin 120Πt T=1/60sec

„ Rectified: E(t)=⏐120 sin 120Πt⏐ T /2 1/ 120 ATytdto ==2[( 1 / ) ( ) 2 / ( 1 / 60 ) 120 sin 120π tdt = 76 . 4 ∫∫0 0

11 T / 2 „ ATytntTdtn = (/)42 ()cos((π )/) ∫ 0

1/ 120 = 4/( 1 / 60 ) 120 sin 120ππtntdt cos 120 ∫ 0 „ For values of n that are odd, the coefficient An is identically zero. „ For values of n that are even, the result is 120 − 2 2 An = ( + ) π n − 1 n + 1

„ The Fourier series for the function |120 sin 120πt| is

„ 76.4 – 50.93 cos 240πt – 10.10 cos 480πt – 4.37 cos 720πt

„ The frequency content of a signal is found by examining the amplitude of various frequency components, and by expanding the fourier series and plotting amplitudes of sine and cosine terms.

12 Fourier Transform

„ In Fourier Series analysis we worked with arbitrary but known signals. The coefficients of the FS specified the amplitude of the sine and cosines having a specific frequency. In practical applications, we may not know anything about the form or frequency of the signal.

„ We need a method to decompose signals in terms of their frequency components.

Fourier Transform

„ From earlier work: ∞ yt()=+ Aon∑ ( A cosnωt +B n sinnωt ) n=1 T / 2 ATytn = (/)2 ()cosnωtdt ∫ −T / 2

T / 2 BTytn = (/)2 ()sinnωtdt ∫ −T / 2 ω = 2πf

Assumption:

„ We remove the constraint that the signal be a periodic function, and let the period go to infinity. „ The coefficients become continuous functions of frequency. ∞ ω = ωtdt A()∫ −∞yt ()cos

∞ B()ω = yt ()sinωtdt ∫ −∞

13 Assumption

„ The fourier coefficients of A(ω) and B(ω) are components of fourier transforms of y(t) Υ()ω =−Aw () iB (ω)

„ Then substituting ∞ Υ(ω )=−yt ( )(cosωt i sinωt )dt ∫ −∞

− iθ „ Using identity ei= cosθ − sinθ ∞ leads to Υ()ω = yte ()− iωt dt (Eq 2.29) ∫ −∞ Substituting for cyclic frequency F=w/2π=1/T (Eq 2.30) ∞ Υ ()fytedt= ()− ift2π ∫ −∞

These are the two-sided fourier transforms of y(t)

„ If we know the amplitude-frequency properties of a signal (fourier transform), the inverse-FT will reconstruct y(t). ∞ ift2π yt()= (12 /π ) Yf ( ) e dt ∫ −∞ „ Y(f) describes the signal as a continuous function of frequency. If y(t) is known or measured, then its fourier transform provides the amplitude-frequency properties of the signal that are not apparent from the time-based signal.

14 „ The fourier transform is a complex number ΥΥ()fAfiBf=− () () = () feifθ ()

„ The magnitude is given by 2 2 ΥΥΙΥ()fff=+ Re[()]M [()] „ Phase shift − 1 θ ()fff= tan(([()])/(Re[()]))ΙΥM Υ

„ *Amplitude spectrum: Cf()=+ Af ()2 Bf ()2 „ *Phase shift: θ ()fBfAf= tan(())/(())− 1 „ Power spectrum: (())/Cf2 2 „ The Fourier transform has provided a method to decompose a measured signal y(t) into its amplitude-frequency components.

Discrete Fourier Transform

„ y(t) is represented by y(rδt) for r=1,2,…N in a discrete time series. y(t) is decided by a set of impulses of amplitude determined by y(t) at nδt. „ An approximation of the fourier transform for discrete data sets is called the discrete fourier transform (DFT) N − irkN2π / Υ ()fNyrteK = 2 /∑ (δ ) r=1 „ For k=1,2,…N/2

15 „ In this development, t was replaced with rδt and f was replaced with k/(Nδt) „ δf = frequency resolution = 1/(Nδt) „ δt = sampling period „ N = number of samples „ k = number of frequency increments to N/2

„ Fk = kδf = a given frequency

Note:

„ Through the DFT a discrete data signal can be decomposed into Amplitude-Frequency content and then can be reconstructed into a fourier series of known functions. „ The typical DFT used in software is the FFT or the fast fourier transform. It has been optimized for algorithm efficiency.

„ Representation of a simple periodic as a discrete signal „ y(t)=10sin2πt created discrete set using time integral 0.125 s

„ f1=w1/2π=1 hz

„ C(f1=1Hz)=10V

„ θ(f1)=0 „ δt=0.125 s „ N=8

16 Figure 2.20

„ Note plot of Amplitude Spectrum

„ Y(f)=0 at all frequencies except f1=1 Hz at which point it is 10, which corresponds to what we know about the signal g(t).

17