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## Lecture 7: Digital Signals

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**SignalsandSpectral Methods**in Geoinformatics Lecture 7: Digital Signals**Digital Signals**1 0 0 1 0 1 0**Digitalization of signals**Transformation of analog signals into digital ones by means of Α. PCM =Pulse Code Modulαtion) Β. Delta modulation**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**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**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)**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**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**Digitalization of signals**m(t) t**Digitalization of signals**Sampling determination of values mn = m(nΔΤ) at intervals ofΔΤ m(t) t**Digitalization of signals**Sampling determination of values mn = m(nΔΤ) at intervals ofΔΤ m(t) t m1 m2 m3 m4 m5 ΔT**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**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**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**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**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**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**-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**-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**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**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**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**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**Signaling Format**Transmission of digital signals**Signaling Format**Transmission of digital signals Binarysignal to be transmitted = sequence{bi} with bi = 0or bi =1**Signaling Format**Transmission of digital signals Binarysignal to be transmitted = sequence{bi} with bi = 0or bi =1 Transmission with new signalm(t)with possible values1, 0, 1**Signaling Format**Transmission of digital signals Binarysignal to be transmitted = sequence{bi} with bi = 0or bi =1 Transmission with new signalm(t)with possible values1, 0, 1 A time intervalδtis assignedto every digit bidivided to 2 equal parts**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 values1, 0, 1 A time intervalδtis assignedto every digit bidivided to 2 equal parts**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 values1, 0, 1 mia mib A time intervalδtis assignedto every digit bidivided to 2 equal parts**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 values1, 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**Transmission of digital signals bi Binarysignal to be transmitted = sequence{bi} with bi = 0or bi =1 Transmission with new signalm(t)with possible values1, 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**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 values1, 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**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 values1, 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**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)**Final transmission with one of the following 3**modulationmodes 1 0 0 1 0 1 NRZ**Final transmission with one of the following 3**modulationmodes 1 0 0 1 0 1 NRZ ASK modulation (Amplitude Shift Keying) ASK**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**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**Modulation:Original signal d(t)with digit length**Tmodulated asy(t) = d(t)cos(ω0t) Spread spectrum technique**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**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**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**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**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**Correlation of digital signals**Digital signal = linear combination of orthogonal pulses**Correlation of digital signals**Digital signal = linear combination of orthogonal pulses Elementaryorthogonal pulse (durationΤ, amplitude 1,centert = 0):