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Lesson 6: Sampling Analog Signals 4/25/2016 Lesson 6: Sampling Analog Signals ET 438b Sequential Control and Data Acquisition Department of Technology Lesson 6_et438b.pptx 1 Learning Objectives After this presentation you will be able to: Identify the steps in sampling an analog signal Indentify the frequency spectrum of a sampled signal Determine the minimum sampling rate of an analog signal Determine if a sampled signal contains aliased signals. Lesson 6_et438b.pptx 2 1 4/25/2016 Sampled Signals Sampling Process Data Acquisition Card Signal Sensor Conditioner Physical parameter Amplify Multiplexer Filter Sample & Linearize ADC Hold Other analog Input channels To computer Data bus Lesson 6_et438b.pptx 3 Sampled Signals-Representation of Signal Analog Signal - defined at every point of independent variable For most physical signals independent variable is time Sampled Signal - Exists at point of measurement. Sampled at equally spaced time points, Ts called sampling time. (1/Ts =fs), sampling frequency Analog Example s(t) 5sin(2250t 60 ) 2cos(2500t 120 ) Sampled Example s(n) 5sin(2250Ts n 60 ) 2cos(2500Ts n 120 ) n=sample number Lesson 6_et438b.pptx 4 2 4/25/2016 Sampled Data Examples Representation of analog signal Ts Representation of sampled analog signal Lesson 6_et438b.pptx 5 Sample and Hold Operation Sample and Hold Circuit Operating Modes tracking = switch closed Control input operates hold= switch open solid –state switch at sampling rate fs Sample and Hold Parameters Acquisition Time - time from instant switch closes until Vi within defined % of input. Determined by input time constant t = RinC 5t value = 99.3% of final value decay rate - rate of discharge of C Impedance when circuit is in hold mode Buffer aperture time - time it takes switch to open. Lesson 6_et438b.pptx 6 3 4/25/2016 Sample and Hold Signals Pulse generator closes switch and captures signal value 10 10 5 s(t) 0 a(t) Amplitude 5 10 10 0 2 4 6 8 10 0 t 10 Time Sampled Signal Analog Signal Analog and sampled signal Pulse generator output Amplitude Modulated s(t)=p(t)∙a(t) Lesson 6_et438b.pptx 7 Sample and Hold Output Sample must be held while digital conversion takes place. Total time to digitize tc = ta + td Where tc = total conversion time ta = total acquisition time td = total digital conversion time Hold Time Lesson 6_et438b.pptx 8 4 4/25/2016 Frequency Spectrum Sampling is modulation. Shifts all signal frequency components and generates harmonics fc=1000 Hz fI1=50 Hz fI2=25 Hz v(t) 1sin(21000t)1sin(250t) 1sin(225t) Carrier Information Modulation produces sums and differences of carrier and information frequencies st fh1= fc±fI1 for the 1 information frequency nd fh2= fc±fI1 for the 2 information frequency fhi= fc±fIi for the i-th information frequency Lesson 6_et438b.pptx 9 Frequency Spectrum Frequency Components fh1= fc±fI1 =1000 Hz ± 50 Hz = 1050 Hz and 950 Hz fh2= fc±fI1 =1000 Hz ± 25 Hz = 1025 Hz and 975 Hz Frequency Spectrum Plot | v | 1000 Hz 1.0 0.5 950 Hz 975 Hz 1025 Hz 1050 Hz Frequency Lesson 6_et438b.pptx 10 5 4/25/2016 Frequency Spectrum Complex signals usually have a frequency spectrum that is wider. Can be visualized with continuous f plot and found with an Fast Fourier Transform (FFT) | v | highest frequency in signal dc f=0 Frequency Frequency spectrum of input signals sample & hold must be known to accurately reproduce original signal from samples Lesson 6_et438b.pptx 11 Nyquist Frequency and Minimum Sampling Rate To accurately reproduce the analog input data with samples the sampling rate, fs, must be twice as high as the highest frequency expected in the input signal. (Two samples per period) This is known as the Nyquist frequency. fs(min) = 2fh Where fh = the highest discernible f component in input signal fs(min) = minimum sampling f Nyquist rate is the minimum frequency and requires an ideal pulse to reconstruct the original signal into an analog value Lesson 6_et438b.pptx 12 6 4/25/2016 Sampled Signal Frequency Spectrum Sampling with fs >2fh Amplitude fs+fh fh fs-fh fs 2fs-fh 2f 2fs+fh Frequency s Sampling at less than 2fh causes aliasing and folding of sampled signals. Folded Frequencies Amplitude fs+fh fh fs-zfh fs 2fs-fh 2f 2fs+fh Frequency s Lesson 6_et438b.pptx 13 Nyquist Frequency and Aliasing Only signals with frequencies below Nyquist frequency will be correctly reproduced Example: Given the following signal, determine the minimum sampling rate (Nyquist frequency) s(t) 2sin(2100t) 5sin(2250t) 1.5cos(2500t) 3sin(2400t) Find the highest frequency component: 100 Hz, 250 Hz, 500 Hz, 400 Hz f = 500 Hz h fs(min) = 2fh fs(min) = 2(500 Hz)= 1000 Hz Lesson 6_et438b.pptx 14 7 4/25/2016 Nyquist Frequency and Aliasing Example: Given the following signal, determine the minimum sampling rate (Nyquist frequency) s(t) 1.5sin(175t) 3sin(250t) 0.5cos(800t) 1.75sin(900t) Convert the radian frequency to frequency in Hz by dividing values by 2 175 250 800 900 f 87.5 Hz f 125 Hz f 400 Hz f 450 Hz 1 2 2 2 3 2 4 2 Find the highest frequency component: 450 Hz fs(min) = 2fh fs(min) = 2(450 Hz)= 900Hz Lesson 6_et438b.pptx 15 Aliased Frequencies Sampling analog signal below 2fh produces false frequencies. Aliased frequencies determined by: falias fI n fs 0 falias fnyquist f f s nyquist 2 Where: fI = sampled information signal with fI>fnyquist fs = sampling frequency (Hz) n = sampling harmonic number falias = aliased frequency fnyquist = one-half sampling frequency Lesson 6_et438b.pptx 16 8 4/25/2016 Samples/Period and Aliasing Correct signal representation requires at least two samples/period fs TI Ns fI Ts fs fI and TI Ts Where Ns = number input signal samples per period of sampling frequency fs = sampling frequency (Hz) fI = highest information signal frequency (Hz) Ts = sampling period, 1/fs, (seconds) TI = period information signal’s highest frequency (1/fI) Lesson 6_et438b.pptx 17 Sampling/Aliasing Examples Example 1: A fs=1000 Hz sampling frequency samples an information signal of fI=100 Hz . Determine samples/period, the resulting recovered signal ,and aliased frequencies if present Determine the number of samples/ period 1000 Hz 0.01 S N 10 samples/period Above Nyquist rate of 2 s 100 Hz 0.001 S f 1000 Hz f s 500 Hz Signals below 500 Hz nyquist 2 2 reproduced without aliasing View the frequency spectrum using FFT of samples Lesson 6_et438b.pptx 18 9 4/25/2016 Sampling/Aliasing Examples 500 Hz Frequency Spectrum 100 Hz Nyquist Recovered Limit 2 100 1.5 1 Amplitude 0.5 0 0 100 200 300 400 500 600 Frequency Frequency Spectrum Lesson 6_et438b.pptx 19 Sampling/Aliasing Examples Example 2: A fs=60 Hz sampling frequency samples an information signal of fI=100 Hz . Determine samples/period, the resulting recovered signal ,and aliased frequencies if present Determine the number of samples/ period 60 Hz 0.01 S N 0.6 samples/period Below Nyquist rate of 2 s 100 Hz 0.001666 S Aliased signals will occur due to low sampling rate f 60 Hz f s 30 Hz Signals below 30 Hz nyquist 2 2 reproduced without aliasing Now compute the aliased frequency for 1st sampling harmonic Lesson 6_et438b.pptx 20 10 4/25/2016 Sampling/Aliasing Examples Alias frequencies for 1st harmonic of sampling f (n=1) falias 100 Hz160 Hz 40 Hz 60 Hz f 30 Hz nyquist 2 0 f 30 Hz alias The falias is outside range 0-30 Hz, (40 Hz > 30 Hz) No recovered signal Find alias frequencies of 2nd sampling harmonic f (n=2) falias 100 Hz 260 Hz 20 Hz The falias in range 0-30 Hz, 20 Hz recovered signal Lesson 6_et438b.pptx 21 Sampling/Aliasing Examples Frequency Spectrum 30 Hz Nyquist 3 Limit 20 Hz 2.5 Alias f falias fI n fs 2 0 falias fnyquist 1.5 Amplitude f f s nyquist 1 2 0.5 0 0 5 10 15 20 25 30 35 Frequency Frequency Spectrum Lesson 6_et438b.pptx 22 11 4/25/2016 Sampling/Aliasing Examples Example 3: A fs=80 Hz sampling frequency samples an information signal of fI=100 Hz . Determine samples/period, the resulting recovered signal ,and aliased frequencies if present Determine the number of samples/ period 80 Hz 0.01 S N 0.8 samples/period Below Nyquist rate of 2 s 100 Hz 0.00125 S Aliased signals will occur due to low sampling rate f 80 Hz f s 40 Hz Signals below 40 Hz nyquist 2 2 reproduced without aliasing Lesson 6_et438b.pptx 23 Sampling/Aliasing Examples Alias frequencies for 1st harmonic of sampling f (n=1) falias 100 Hz180 Hz 20 Hz 60 Hz ffalias fI n fs 40 Hz nyquist 2 0 f f 0 falias 40nyquist Hz alias fs fThe falias is inside range 0-40 Hz 20nyquist Hz recovered2 signal Find alias frequencies of 2nd sampling harmonic f (n=2) falias 100 Hz 280 Hz 60 Hz The falias outside range 0-40 Hz, No recovered signal Lesson 6_et438b.pptx 24 12 4/25/2016 Sampling/Aliasing Examples Frequency Spectrum 40 Hz Nyquist Limit 20 Hz 3 Alias f 2.5 2 1.5 Amplitude 1 0.5 0 0 10 20 30 40 50 Frequency Frequency Spectrum Lesson 6_et438b.pptx 25 Sampling/Aliasing Examples Example 4: A fs=100 Hz sampling frequency samples an information signal of fI=100 Hz . Determine samples/period, the resulting recovered signal ,and aliased frequencies if present Determine the number of samples/ period 100 Hz 0.01 S N 1 samples/period Below Nyquist rate of 2 s 100 Hz 0.01 S Aliased signals will occur due to low sampling rate f 100 Hz f s 50 Hz Signals below 50 Hz nyquist 2 2 reproduced without aliasing Lesson 6_et438b.pptx 26 13 4/25/2016 Sampling and Aliasing Examples Alias frequencies for 1st harmonic of sampling f (n=1) falias 100 Hz1100 Hz 0 Hz 100 Hz f 50 Hz nyquist 2 0 f 50 Hz alias The falias is inside range 0-50 Hz.
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