Signal Processing Level IV/V CITE, BTS/DUT/Licence. Modulation FSK à phase continue i_8 Tension de commande V1 Interrupt SCOPE AD-DA EVENT Fs = 1E5 Hz 1:1.15 i_4 Modulation FSK à phase continue LOWPASS 2nd order 4 w =43rad/s xi =.5; G =2 dt i_2 i_3 MUX true freq G_SIN 1. Hz in i_5 5 in2 mux cmd gsin phi mul Filtre passe-bas déphase -9 à 1kHz.2 dt1 i_ i_1 bauds message in 1 2 LOWPASS 2nd order 6 w =6rad/s xi =.8; G =3 lp2a1 f 7 t i_7 T F gt COMPARE in ref comare MESSAGE in Filtre passe-bas SPECTRUM 124point FFT in half spec db win: Blackman_Harris 1 clk i_6 in1 lp2a 3 false Tension de commande V2 Démodulation FSK à quadrature.45 dt2 Comparateur mesage frtool Topics and Reports Author : N Gally KOMA Professor in Electronic BTS Z.A. La Clef St Pierre - 5, rue du Groupe Manoukian 7899 ELANCOURT France Tel. : 33 ()1 3 66 8 88 - Fax : 33 ()1 3 66 72 2 e-mail : ge@didalab.fr - Web : www.didalab.fr Ref : ETD 41 41
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SUMMARY EX.1 : DIGITAL FILTERS... 5 1.1 REMIND ON DIGITAL SYSTEMS... 5 1.2 DIGITAL FILTERS... 9 EX.2 : RECURSIVE FILTERS... 13 2.1 EXAMPLES OF 4 NON-RECURSIVE FILTERS... 13 EX.3 : RECURSIVE FILTER: TRAPEZIUM METHOD... 23 3.1 RECURSIVE LOW-PASS FILTER OF 1 ST ORDER... 23 3.2 RECURSIVE HIGH-PASS FILTER OF 1 ST ORDER... 24 3.3 RECURSIVE LOW-PASS FILTER OF 2 ND ORDER... 24 3.4 PRACTICAL WORKS... 25 EX.4 : SAMPLE: NYQUIST -SHANON THEOREM... 38 4.1 TARGET... 38 4.2 SAMPLE PRINCIPLE DIAGRAM... 38 4.3 FUNCTION... 38 4.4 EXPRESSION OF SAMPLED SIGNAL S(T)... 41 4.5 COMPLEX WRITING OF FOURIER SERIES: BI-LATERAL SERIES... 42 4.7 PRACTICAL WORKS... 46 EX.5 : TRANSMISSION IN BASEBAND: SPACTRALE DENSITY... 58 5.1 TARGET... 58 5.2 DEFINITIONS... 58 5.3 AMPLITUDE SPECTRA... 6 5.4 POWER SPECTRAL DENSITY... 65 5.5 PRACTICAL WORKS... 67 EX.6 : ASK DIGITAL MODULATION... 78 6.1 INTRODUCTION... 78 6.2 TYPES OF MODULATION... 78 6.3 ASK DIGITAL MODULATION... 79 6.4 PRACTICAL WORKS... 81 EX.7 : PSK DIGITAL MODULATION... 88 7.1 MODULATION... 88 7.2 DEMODULATION... 89 7.3 PRACTICAL WORKS... 89 EX.8 : FSK DIGITAL MODULATION... 94 8.1 FM SIGNAL DEFINATION... 94 8.2 FSK MODULATION WITH DISCONTINOUS PHASE... 95 8.3 FSK MODULATION WITH CONTINOUS PHASE... 96 8.4 FSK DEMODULATION... 97 8.5 PRACTICAL WORKS... 97 EX.9 : CONTINUOUS ANALOG MODULATION: AM... 11 9.1 DEFINTION... 11 9.2 MODULATION PURPOSE... 11 9.3 PRINCIPLE DIAGRAMS OF A MAPS... 11 9.4 DEMODULATION OF MAPS OR DSB-SC... 111 9.5 SINGLE SIDE BAND MODUDLATION: BLU OR SSB... 112 9.6 DEMODULATION OF AN AM MAPS WITH CARRIER RECONSTITUTION... 115 Page : 3 / 15
9.7 PRACTICAL WORKS... 116 EX.1 : CONTINUOUS ANALOG MODULATION: FM... 136 1.1 DEFINITION... 136 1.2 MODULATION BY A SINUSOIDAL SIGNAL... 136 1.3 FM MODULATION... 138 1.4 DEMODULATION... 138 1.5 PRACTICAL WORKS... 141 1.6 FM MODULATOR-DEMODULATOR DIAGRAM... 142 Page : 4 / 15
Ex.4 : SAMPLE: NYQUIST -SHANON THEOREM 4.1 TARGET At usually regular instants, pick up the values taken by a signal for various types of treatment; examples: 1- Wired or radio transmission, time-division multiplexing, and analog processing, 2- Digital processing (optionally followed by a restitution of the analog signal) needs to constantly maintain the value of the samples during their conversion in digital information; this requires functions : Sample/Hold, Analog-digital conversion ADC, Digital processing : computer or DSP Digital- analog conversion if necessary. In any case, it must be ensured that the processed signal is not substantially altered by sampling operation: limited information loss no change in the spectrum, that is to say, no added information 4.2 SAMPLE PRINCIPLE DIAGRAM The diagram of the sampling function is given as figure 1. h(t) e(t) s(t) K R e(t) : input signal. h(t) : sampling signal. The rectangular signal h (t) is defined as: t t s(t) : output signal K : switch controlled by h (t) assumed to be perfect. 4.3 FUNCTION The h(t) signal has 2 states : a high state and a low state. a) h(t) is in the state (for example high state), K is closed so s(t) = e(t) Page : 38 / 15
b) h(t) is in another state (low state), K is opened, s(t) =. Consider that h(t) as an amplitude 1 or signal. The operation may be assimilated to a multiplication operation see EX modulation: K Unit: [V] If the amplitude of h(t) is 1, then the module k will be 1. The expression for s (t) is determined from the decomposition in h (t) Fourier series. If h(t) is a periodic signal of T e or T s period (figure 2), it may be decomposed in Fourier series. w w w The signal h (t) can be formed as: w j with : j a QUESTION : Determine the coefficients expression of Fourier series; ANSWER For n ranging from to infinity, the coefficients of the series are calculated as follows: w t a a w Page : 39 / 15
t w t w w t w t w w w w t w w t w This gives for the coefficients, the following expressions: p t ) = pa p t pa p A n is the amplitude of the spectral lines of h(t). a) If a is equal to.5, the coefficients a n are null. Only the b n coefficients remain: the fundamental F s and the impaired harmonics of the form nf s = (2p)F s ; their amplitude A 2p is given by the relation : p b) If a is any fundamental and all harmonics are present and are for the amplitude, the An is : pa pa pa pa pa This gives: pa p pa ap Page : 4 / 15
By multiplying by a,, we obtain ; a p ap p ap p ap ap a ap ap Example : amplitude h(t) = 1V. A n Coefficients in terms of a for n = 1 a,1,2,3,4,5,6,7,8,9 a ap ap,1967,3742,515,655,6366,655,515,3742,1967 A n Coefficients in terms of n for a = 1 n 1 2 3 4 5 6 7 8 9 a ap ap,655,1871,1247,1514,,19,535,468,673 Remark: The amplitude of the lines is zero for na = k integer: sin (nap) = sin (kp) = 4.4 EXPRESSION OF SAMPLED SIGNAL s(t) To simplify the calculation, we suppose that e(t) is a pure sinusoidal signal with the pulsation w (and the frequency f = w/2p). From the superposition theorem, the calculation can be applied to any signal. w p Page : 41 / 15
QUESTION 1- Determine the expression of the sampled signal 2- Show that the spectrum is composed of the frequency f and an infinite number of frequency lines nf s ± f for n ranging from 1 to infinity. ANSWER 1- With : w p w w j w w j a w w w j By w = 2pf and w s = 2pf s, we obtain: a p p p j p p j a p p j p j 2- We note that the spectrum is composed: 1- Of the frequency f 2- Of the frequencies - and for n ranging from 1 to infinity. 4.5 COMPLEX WRITING OF FOURIER SERIES: BI-LATERAL SERIES The Fourier series term of a periodic function f(t), (see the previous), of T period and of w pulsation is : w w Let: w w By expressing w w in their complex form, we obtain: w w Page : 42 / 15 w w
QUESTION 1- Determine the expression of the complex coefficients of the Fourier series: C n and C -n = (conjugate complex of C n ). 2- Find the modules of C n and C -n in terms of A n coefficients of the unilateral series. ANSWER 1- By multiplying the numerator and denominator with the term of the second member by j and putting the exponential as factor, we obtain : Let: w w with: So it shows the relation of u n : This gives the relation for f(t): w w w w This expression can be written in another form by make the n range from - to + : w Retake the previous Fourier series coefficients calculation, but this time the series is under the exponential form. We obtain: w w w w w w w w w w Page : 43 / 15
The operation w w = leads: w w w w In the same way, the operation = results in: w In summary, we will have: w w w 2- Calculate the modules of bilateral spectrum elements of the f(t) function. Page : 44 / 15
Important remark It is found that the amplitude of the bilateral spectrum components is half of that of unilateral spectrum. Page : 45 / 15
4.7 PRACTICAL WORKS 4.7.1 Manipulation Purpose: The purpose of the manipulation is to demonstrate the Nyquist-Shannon theorem. QUESTION State the Nyquist-Shannon theorem. ANSWER Nyquist-Shannon theorem We don t lose the information by reconstructing a signal from its samples if the sampling frequency is at least equal to twice the highest contained frequency in the spectrum of the sampled signal. 4.7.2 Application diagram Realize the application diagram as figure 2. The figure 2 indicates the information defining the sampling parameters. The figures 3 to 9 can demonstrate the effected calculations in the theoretical part. 4.7.3 Nyquist-Shannon theorem demonstration The figures 12 to 15 can demonstrate the recovery or folding of sampled signal thanks to the module edt1. The figure 16.allowed to make the message spectrum vary by acting on the module dt1 by means of the mouse ; observe at the same time the spectrum cover on the figure 16.b. Page : 46 / 15
WAIT AD-DA SAMPLE Fs =1e5Hz 6 SCOPE 1:1 freq 1 G_SIN 3 Hz gsin phi 2 in.5 1.5 gof in 5 SWITCH switch cmd 3.6 dt duty freq 3 G_RECT 2 Hz grect 4 in 2 frtool Nyquist_shannon_1.fib fig.2 e(t) gof t h(t) Ts frtool T t F s(t) switch 1 2 3 4 5 6 µs fig.3 Page : 47 / 15
gof Message Signal mul_in2 fe or fs sampler Signal mul Sampled Message Signal 1 2 3 4 5 fig.4 ms Message Signal: pure sine wave gof.25.999 1. 1. mul_in2.999 mul.25 db -2 specan -4 6.5 6. -6-8 2 4 32kHz 4kHz 1 2kHz 3 4 fig.5 khz Spectrum of the sampled message: fs = 2kHz fm = 4kHz Page : 48 / 15
gof.789.847 1. mul_in2. mul.789. db specan -2-4 -21.9-22.1-6 -8 2 4 16kHz 1 16kHz 2 24kHz 3 4 fig.6 khz gof.581.994 1. 1. mul_in2 mul.. db -2 specan -4 6. -24.7-6 -8 2 4 4kHz 1 2kHz 3 4 4kHz fig.7 khz Spectrum : fm, fs-fm, n.fs, fs+fm, repeated every fs : 2kHz, 4 kkhz Page : 49 / 15
gof.977.264 1. mul_in2..977 mul. db specan -2-4 -3.7-6 -8 2 4 16kHz 1 2 3 36kHz 4-3.9 44kHz fig.8 khz gof.977.31 mul_in2...977 mul. db specan -2-4 -3.7-24.7-6 -8 2 4 8kHz 1 2 3 36kHz 4 4kHz fig.9 khz Page : 5 / 15
Spectrum of a sampled or sampled/blocked signal freq i_ G_SIN 1 Hz.4 dt duty freq i_8 G_RECT 2 Hz 1 in i_9 h(t) freq freq freq gsin phi i_1 G_SIN 2 Hz gsin1 phi i_2 G_SIN 3 Hz gsin2 phi i_3 G_SIN 4 Hz in1 in2 in3 i_5.1.15.2 wsum3 in1 in2 in3 grect i_6 1..15.35 wsum31 frtool i_7 in.5.5 gof 2 in sample i_1 S & H samold e(t) in cmd i_11 3 SWITCH switch i_12 in 1 4 gain s(t) seb(t) freq gsin3 phi i_4 G_SIN 5 Hz Interrupt AD-DA EVENT Fs = 1e5 Hz i_13 SCOPE 1:1 1 SPECTRUM 124point FFT 5 in half spec db win: Blackman_Harris gsin4 phi Occupation_Spectrale.fib fig.1 Page : 51 / 15
gof.4.322 mul_in2.. mul.4. Vaverage fmin fmax specan db -2-4 -6-8 2 4-44.4 8kHz -27.8 Ideal low-pass filter 1.7kHz 5.8kHz 1 2 3 4 5 khz fig.11 Spectrum of the sampled message signal : fmin = 1kHz ; fmax = 5kHz ; aliasing but no spectrum recovery Page : 52 / 15
Spectral_recovery.fib fig.12 Page : 53 / 15
gof.187.274 1. 1. mul_in2 Ideal low-pass mul.187.274 specan db -2-4 -6-8 2 4 fmax -25.6 8kHz -31. fs-fmax 8kHz 1 12kHz 2 3 4 5 khz fig.13 fmax =8kHz è fmax < fp-fmax è fmax < fp-fmax ; aliasing but no spectrum overlap. Recoverable Message Page : 54 / 15
grect gof switch gain specan db -2-4 -6-8 2 4 Absence of the sampling signal ray fs of 2kHz and of its 2kHz harmonics 1 2 3 4 5 khz Spectrum of sample-hold signal fig.14 Page : 55 / 15
gof.521.647 1. 1. mul_in2 Message Spectrum Message folded spectrum mul.521.647 specan db -2-4 -6-8 2 4-31.3 11.9kHz -25.7 Ideal low-pass filter : Recovery of the message and the part of folded spectrum 7.3kHz 1 13.kHz 2 3 4 5 khz fig.15 fmax = 13Khz è fmax > fs-fmax = 7kHzè therefore spectrum recovery; the low-pass filter also recovers a part of the folded spectrum Page : 56 / 15
fig.16.a fig.16.b fig.16 Page : 57 / 15