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Lecture 7: Digital Signals

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# Lecture 7: Digital Signals

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## Lecture 7: Digital Signals

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1. SignalsandSpectral Methods in Geoinformatics Lecture 7: Digital Signals

2. Digital Signals 1 0 0 1 0 1 0

3. Digitalization of signals

4. Digitalization of signals Transformation of analog signals into digital ones by means of Α. PCM =Pulse Code Modulαtion) Β. Delta modulation

5. Digitalization of signals Transformation of analog signals into digital ones by means of Α. PCM =Pulse Code Modulαtion) Β. Delta modulation A1. Sampling   Α2. Quantization   Α3. Codification

6. Digitalization of signals Transformation of analog signals into digital ones by means of Α. PCM =Pulse Code Modulαtion) Β. Delta modulation A1. Sampling   Α2. Quantization   Α3. Codification Sampling theorem

7. Digitalization of signals Transformation of analog signals into digital ones by means of Α. PCM =Pulse Code Modulαtion) Β. Delta modulation A1. Sampling   Α2. Quantization   Α3. Codification Sampling theorem Ifm(t)is a band-limited signal(M(ω) = 0 for|ω| > ωΜ) then the signal m(t)can be reconstructed from sampling values (at equal distancesΔT)

8. Digitalization of signals Transformation of analog signals into digital ones by means of Α. PCM =Pulse Code Modulαtion) Β. Delta modulation A1. Sampling   Α2. Quantization   Α3. Codification Sampling theorem Ifm(t)is a band-limited signal(M(ω) = 0 for|ω| > ωΜ) then the signal m(t)can be reconstructed from sampling values (at equal distancesΔT) provided that the sampling is dense enough, specifically when

9. Digitalization of signals Transformation of analog signals into digital ones by means of Α. PCM =Pulse Code Modulαtion) Β. Delta modulation A1. Sampling   Α2. Quantization   Α3. Codification Sampling theorem Ifm(t)is a band-limited signal(M(ω) = 0 for|ω| > ωΜ) then the signal m(t)can be reconstructed from sampling values (at equal distancesΔT) provided that the sampling is dense enough, specifically when The signal is reconstructed through the relation

10. Digitalization of signals Sampling determination of values mn = m(nΔΤ) at intervals ofΔΤ m(t) t

11. Digitalization of signals Sampling determination of values mn = m(nΔΤ) at intervals ofΔΤ m(t) t m1 m2 m3 m4 m5 ΔT

12. Digitalization of signals Sampling determination of values mn = m(nΔΤ) at intervals ofΔΤ m(t) t m1 m2 m3 m4 m5 ΔT initialvaluexk -0.96 -2.33 -1.82 0.14 2.43

13. Digitalization of signals Sampling determination of values mn = m(nΔΤ) at intervals ofΔΤ m(t) Quantization replacement of each value mn = m(nΔΤ) with the closestvaluexk from a predefined discrete set ..., xi, xi+1, ..., xi+n, ... (usually equidistant) t m1 m2 m3 m4 m5 ΔT initialvaluexk -0.96 -2.33 -1.82 0.14 2.43

14. Digitalization of signals Sampling determination of values mn = m(nΔΤ) at intervals ofΔΤ m(t) Quantization replacement of each value mn = m(nΔΤ) with the closestvaluexk from a predefined discrete set ..., xi, xi+1, ..., xi+n, ... (usually equidistant) t m1 m2 m3 m4 m5 ΔT initialvaluexk -0.96 -2.33 -1.82 0.14 2.43

15. Digitalization of signals Sampling determination of values mn = m(nΔΤ) at intervals ofΔΤ 4 m(t) 3 Quantization replacement of each value mn = m(nΔΤ) with the closestvaluexk from a predefined discrete set ..., xi, xi+1, ..., xi+n, ... (usually equidistant) 2 1 0 -1 -2 -3 t m1 m2 m3 m4 m5 ΔT initialvaluexk -0.96 -2.33 -1.82 0.14 2.43

16. Digitalization of signals Sampling determination of values mn = m(nΔΤ) at intervals ofΔΤ 4 m(t) 3 Quantization replacement of each value mn = m(nΔΤ) with the closestvaluexk from a predefined discrete set ..., xi, xi+1, ..., xi+n, ... (usually equidistant) 2 1 0 -1 -2 -3 t m1 m2 m3 m4 m5 ΔT initialvaluexk -0.96 -2.33 -1.82 0.14 2.43 quantumvalue

17. Digitalization of signals Sampling determination of values mn = m(nΔΤ) at intervals ofΔΤ 4 m(t) 3 Quantization replacement of each value mn = m(nΔΤ) with the closestvaluexk from a predefined discrete set ..., xi, xi+1, ..., xi+n, ... (usually equidistant) 2 1 0 -1 -2 -3 t m1 m2 m3 m4 m5 ΔT initialvaluexk -0.96 -2.33 -1.82 0.14 2.43 quantumvalue

18. -1 -2 -2 0 2 Digitalization of signals Sampling determination of values mn = m(nΔΤ) at intervals ofΔΤ 4 m(t) 3 Quantization replacement of each value mn = m(nΔΤ) with the closestvaluexk from a predefined discrete set ..., xi, xi+1, ..., xi+n, ... (usually equidistant) 2 1 0 -1 -2 -3 t m1 m2 m3 m4 m5 ΔT initialvaluexk -0.96 -2.33 -1.82 0.14 2.43 quantumvalue

19. -1 -2 -2 0 2 Digitalization of signals Sampling determination of values mn = m(nΔΤ) at intervals ofΔΤ 4 m(t) 3 Quantization replacement of each value mn = m(nΔΤ) with the closestvaluexk from a predefined discrete set ..., xi, xi+1, ..., xi+n, ... (usually equidistant) 2 1 0 -1 -2 -3 t Codification replacement of the valuexk with a code, i.e. an integerk expressed in the binary system (only digits 0 and 1) m1 m2 m3 m4 m5 ΔT initialvaluexk -0.96 -2.33 -1.82 0.14 2.43 quantumvalue

20. 7 6 5 4 3 2 1 0 -1 -2 -2 0 2 Digitalization of signals Sampling determination of values mn = m(nΔΤ) at intervals ofΔΤ 4 m(t) 3 Quantization replacement of each value mn = m(nΔΤ) with the closestvaluexk from a predefined discrete set ..., xi, xi+1, ..., xi+n, ... (usually equidistant) 2 1 0 -1 -2 -3 t Codification replacement of the valuexk with a code, i.e. an integerk expressed in the binary system (only digits 0 and 1) m1 m2 m3 m4 m5 ΔT initialvaluexk -0.96 -2.33 -1.82 0.14 2.43 quantumvalue code

21. 7 6 5 4 3 2 1 0 -1 -2 -2 0 2 2 1 1 3 5 Digitalization of signals Sampling determination of values mn = m(nΔΤ) at intervals ofΔΤ 4 m(t) 3 Quantization replacement of each value mn = m(nΔΤ) with the closestvaluexk from a predefined discrete set ..., xi, xi+1, ..., xi+n, ... (usually equidistant) 2 1 0 -1 -2 -3 t Codification replacement of the valuexk with a code, i.e. an integerk expressed in the binary system (only digits 0 and 1) m1 m2 m3 m4 m5 ΔT initialvaluexk -0.96 -2.33 -1.82 0.14 2.43 quantumvalue code

22. 7 111 110 6 101 5 100 4 011 3 010 2 001 1 000 0 -1 -2 -2 0 2 2 1 1 3 5 Digitalization of signals Sampling determination of values mn = m(nΔΤ) at intervals ofΔΤ 4 m(t) 3 Quantization replacement of each value mn = m(nΔΤ) with the closestvaluexk from a predefined discrete set ..., xi, xi+1, ..., xi+n, ... (usually equidistant) 2 1 0 -1 -2 -3 t Codification replacement of the valuexk with a code, i.e. an integerk expressed in the binary system (only digits 0 and 1) m1 m2 m3 m4 m5 ΔT initialvaluexk -0.96 -2.33 -1.82 0.14 2.43 quantumvalue code binary code

23. 7 111 110 6 101 5 100 4 011 3 010 2 001 1 000 0 -1 -2 -2 0 2 2 1 1 3 5 010 001 001 011 101 Digitalization of signals Sampling determination of values mn = m(nΔΤ) at intervals ofΔΤ 4 m(t) 3 Quantization replacement of each value mn = m(nΔΤ) with the closestvaluexk from a predefined discrete set ..., xi, xi+1, ..., xi+n, ... (usually equidistant) 2 1 0 -1 -2 -3 t Codification replacement of the valuexk with a code, i.e. an integerk expressed in the binary system (only digits 0 and 1) m1 m2 m3 m4 m5 ΔT initialvaluexk -0.96 -2.33 -1.82 0.14 2.43 quantumvalue code binary code

24. Signaling Format

25. Signaling Format Transmission of digital signals

26. Signaling Format Transmission of digital signals Binarysignal to be transmitted = sequence{bi} with bi = 0or bi =1

27. Signaling Format Transmission of digital signals Binarysignal to be transmitted = sequence{bi} with bi = 0or bi =1 Transmission with new signalm(t)with possible values1, 0, 1

28. Signaling Format Transmission of digital signals Binarysignal to be transmitted = sequence{bi} with bi = 0or bi =1 Transmission with new signalm(t)with possible values1, 0, 1 A time intervalδtis assignedto every digit bidivided to 2 equal parts

29. Signaling Format Transmission of digital signals bi Binarysignal to be transmitted = sequence{bi} with bi = 0or bi =1 Transmission with new signalm(t)with possible values1, 0, 1 A time intervalδtis assignedto every digit bidivided to 2 equal parts

30. Signaling Format Transmission of digital signals bi Binarysignal to be transmitted = sequence{bi} with bi = 0or bi =1 Transmission with new signalm(t)with possible values1, 0, 1 mia mib A time intervalδtis assignedto every digit bidivided to 2 equal parts

31. Signaling Format Transmission of digital signals bi Binarysignal to be transmitted = sequence{bi} with bi = 0or bi =1 Transmission with new signalm(t)with possible values1, 0, 1 mia mib A time intervalδtis assignedto every digit bidivided to 2 equal parts m(t)has values mia and mib(out of–1,0,1)in the 1stand 2ndhalf of the intervalδt, respectively bi = 0  [m0a, m0b] καιbi = 1  [m1a, m1b]

32. Signaling Format Transmission of digital signals bi Binarysignal to be transmitted = sequence{bi} with bi = 0or bi =1 Transmission with new signalm(t)with possible values1, 0, 1 mia mib A time intervalδtis assignedto every digit bidivided to 2 equal parts m(t)has values mia and mib(out of–1,0,1)in the 1stand 2ndhalf of the intervalδt, respectively bi = 0  [m0a, m0b] καιbi = 1  [m1a, m1b] Signaling format = process of transforming the sequence {bi} to the sequence {mia, mib} The values (-1, 0 or 1)ofm0a, m0b, m1a, m1bcompletely define thesignaling format

33. Signaling Format Transmission of digital signals bi Binarysignal to be transmitted = sequence{bi} with bi = 0or bi =1 Transmission with new signalm(t)with possible values1, 0, 1 mia mib A time intervalδtis assignedto every digit bidivided to 2 equal parts m(t)has values mia and mib(out of–1,0,1)in the 1stand 2ndhalf of the intervalδt, respectively bi = 0  [m0a, m0b] καιbi = 1  [m1a, m1b] Signaling format = process of transforming the sequence {bi} to the sequence {mia, mib} The values (-1, 0 or 1)ofm0a, m0b, m1a, m1bcompletely define thesignaling format Example: bi 1 0 1 1 0 0 0 1 m(t) m1a m1b m0a m0b

34. Signaling Format Transmission of digital signals bi Binarysignal to be transmitted = sequence{bi} with bi = 0or bi =1 Transmission with new signalm(t)with possible values1, 0, 1 mia mib A time intervalδtis assignedto every digit bidivided to 2 equal parts m(t)has values mia and mib(out of–1,0,1)in the 1stand 2ndhalf of the intervalδt, respectively bi = 0  [m0a, m0b] καιbi = 1  [m1a, m1b] Signaling format = process of transforming the sequence {bi} to the sequence {mia, mib} The values (-1, 0 or 1)ofm0a, m0b, m1a, m1bcompletely define thesignaling format Example: bi 1 0 1 1 0 0 0 1 m(t) Signaling format: m0a = -1, m0b = 1, m1a = 1, m1b = -1 m1a m1b m0a m0b

35. Signaling formats m1a m1b m0a m0b 1 0 1 1 0 0 0 1 1 1 0 0 (NRZ = Νon Return to Zero) Unipolar NRZ 1 1 -1 -1 Bipolar NRZ GPS ! 1 0 0 0 (RZ = Return to Zero) Unipolar RZ 1 0 -1 0 Bipolar RZ 1 0 0 0 AMI = = Alternate Mark Inversion AMI -1 0 0 0 1 -1 -1 1 Split-Phase (Manchester) Split-Phase (Manchester)

36. Final transmission with one of the following 3 modulationmodes 1 0 0 1 0 1 NRZ

37. Final transmission with one of the following 3 modulationmodes 1 0 0 1 0 1 NRZ ASK modulation (Amplitude Shift Keying) ASK

38. Final transmission with one of the following 3 modulationmodes 1 0 0 1 0 1 NRZ ASK modulation (Amplitude Shift Keying) ASK FSK modulation (Frequency Shift Keying) FSK

39. Final transmission with one of the following 3 modulationmodes 1 0 0 1 0 1 NRZ ASK modulation (Amplitude Shift Keying) ASK FSK modulation (Frequency Shift Keying) FSK PSK modulation (Phase Shift Keying)GPS! PSK

40. Spread spectrum technique

41. Modulation:Original signal d(t)with digit length Tmodulated asy(t) = d(t)cos(ω0t) Spread spectrum technique

42. Modulation:Original signal d(t)with digit length Tmodulated asy(t) = d(t)cos(ω0t) Coding:Multiplication with signal g(t) = ±1with amplitudeA =1and digit length Tg << T z(t) = g(t)d(t)cos(ω0t)(transmitted coded signal) Comprehensible only to those knowing the PRN code g(t) Spread spectrum technique

43. Modulation:Original signal d(t)with digit length Tmodulated asy(t) = d(t)cos(ω0t) Coding:Multiplication with signal g(t) = ±1with amplitudeA =1and digit length Tg << T z(t) = g(t)d(t)cos(ω0t)(transmitted coded signal) Comprehensible only to those knowing the PRN code g(t) Decoding:Multiplication of received signalz(t)with theknown codeg(t) g(t) z(t) = g(t)2 d(t)cos(ω0t) = d(t)cos(ω0t) sinceg(t)2 = (1)2 = 1 : recovery ofmodulated signal without the code Spread spectrum technique

44. Modulation:Original signal d(t)with digit length Tmodulated asy(t) = d(t)cos(ω0t) Coding:Multiplication with signal g(t) = ±1with amplitudeA =1and digit length Tg << T z(t) = g(t)d(t)cos(ω0t)(transmitted coded signal) Comprehensible only to those knowing the PRN code g(t) Decoding:Multiplication of received signalz(t)with theknown codeg(t) g(t) z(t) = g(t)2 d(t)cos(ω0t) = d(t)cos(ω0t) sinceg(t)2 = (1)2 = 1 : recovery ofmodulated signal without the code Demodulation:y(t) = d(t)cos(ω0t)  d(t) = recovery of originalaignal Spread spectrum technique

45. Modulation:Original signal d(t)with digit length Tmodulated asy(t) = d(t)cos(ω0t) Coding:Multiplication with signal g(t) = ±1with amplitudeA =1and digit length Tg << T z(t) = g(t)d(t)cos(ω0t)(transmitted coded signal) Comprehensible only to those knowing the PRN code g(t) Decoding:Multiplication of received signalz(t)with theknown codeg(t) g(t) z(t) = g(t)2 d(t)cos(ω0t) = d(t)cos(ω0t) sinceg(t)2 = (1)2 = 1 : recovery ofmodulated signal without the code Demodulation:y(t) = d(t)cos(ω0t)  d(t) = recovery of originalaignal Spread spectrum technique Bandwidth:from2/Τ iny(t)becomes2/Τg in z(t)Tg << T 2/Tg >> 2/T

46. Modulation:Original signal d(t)with digit length Tmodulated asy(t) = d(t)cos(ω0t) Coding:Multiplication with signal g(t) = ±1with amplitudeA =1and digit length Tg << T z(t) = g(t)d(t)cos(ω0t)(transmitted coded signal) Comprehensible only to those knowing the PRN code g(t) Decoding:Multiplication of received signalz(t)with theknown codeg(t) g(t) z(t) = g(t)2 d(t)cos(ω0t) = d(t)cos(ω0t) sinceg(t)2 = (1)2 = 1 : recovery ofmodulated signal without the code Demodulation:y(t) = d(t)cos(ω0t)  d(t) = recovery of originalaignal Spread spectrum technique Bandwidth:from2/Τ iny(t)becomes2/Τg in z(t)Tg << T 2/Tg >> 2/T spreadspectrum ! Applications:Police communications, GPS

47. Correlation of digital signals

48. Correlation of digital signals Digital signal = linear combination of orthogonal pulses

49. Correlation of digital signals Digital signal = linear combination of orthogonal pulses Elementaryorthogonal pulse (durationΤ, amplitude 1,centert = 0):