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Lecture 7: Digital Signals

Signals and Spectral Methods in Geoinformatics. Lecture 7: Digital Signals. Digital Signals. 1. 0. 0. 1. 0. 1. 0. Digitalization of signals. Digitalization of signals. Transformation of analog signals into digital ones by means of Α. PCM =Pulse Code Modulαtion)

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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 m(t) t

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

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

  13. 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

  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 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

  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

  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. 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 m1 m2 m3 m4 m5 ΔT initialvaluexk -0.96 -2.33 -1.82 0.14 2.43 quantumvalue

  20. -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

  21. 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

  22. 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

  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 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. 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

  25. Signaling Format

  26. Signaling Format Transmission of digital signals

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

  29. 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

  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 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

  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]

  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

  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) m1a m1b m0a m0b

  35. 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

  36. 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)

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

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

  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

  40. 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

  41. Spread spectrum technique

  42. Modulation:Original signal d(t)with digit length Tmodulated asy(t) = d(t)cos(ω0t) 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) 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 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

  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

  47. 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

  48. Correlation of digital signals

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

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

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