Digital Modulation Techniques

Digital Modulation Introduction

• Digital modulation varies the carrier's amplitude, frequency, and phase in proportion to the information signal
• Amplitude Shift Keying (ASK) is produced if amplitude is varied proportional to a digital signal
• Frequency Shift Keying (FSK) is produced if frequency is varied proportionally to a digital signal
• Phase Shift Keying (PSK) is produced if phase is varied proportionally to a digital signal
• Quadrature Amplitude Modulation (QAM) is produced if both amplitude and phase are varied proportionally to a digital signal
• Equation (2.1) depicts digital modulation

System Components

• The precoder in the transmitter performs level conversion and encodes incoming data into groups of bits to modulate an analog carrier
• Modulated carrier signals are shaped (filtered) and amplified
• They are then transmitted through a transmission medium to the receiver
• Transmission media can be metallic cables, optical fiber cables, the Earth's atmosphere, or any combination
• At the receiver, incoming signals are filtered, amplified, and applied to demodulator and decoder circuits
• These circuits extract the original source information from the modulated carrier
• Clock and carrier recovery circuits recover the analog carrier and digital timing signals for demodulation

Information Capacity

• Information capacity is a linear function of bandwidth and transmission time
• I∝B x t: Information capacity (bits per second) is proportional to bandwidth (hertz) and transmission time (seconds)
• If either bandwidth or transmission time changes, a proportional change occurs in information capacity
• The higher the signal-to-noise ratio, the better the performance and the higher the information capacity
• Shannon's Limit is I = Blog2(1 + S/N) or I = 3.32B log10(1 + S/N) where I is information capacity (bps), B is bandwidth (hertz), and S/N is the signal-to-noise power ratio (unitless)
• For a standard telephone circuit with a signal-to-noise power ratio of 1000 (30 dB) and a bandwidth of 2.7 kHz, the Shannon limit for information capacity is 26.9 kbps
• A 26.9 kbps propagation through a 2.7 kHz communication channel may require each transmitted symbol to contain more than one bit avoiding use of a binary system

M-ary Encoding

• M-ary encoding derives its name from the word binary
• M represents a digit that corresponds to the number of conditions, levels, or combinations possible for a given number of binary variables
• A digital signal with 4 possible conditions is an M-ary system where M=4
• Mathematically, N = log2 M, where N is the number of bits necessary and M is the number of conditions, levels, or combinations
• The number of conditions possible with N bits is expressed as 2^N = M

Baud and Bandwidth

• Baud refers to a signal's rate of change on the transmission medium after encoding and modulation and is a unit of transmission, modulation, or symbol rate
• Baud is mathematically reciprocal to the time of one output signaling element
• A signaling element may represent several information bits: baud = 1/ts, where baud is the symbol rate and ts is the time of one signaling element
• The minimum theoretical bandwidth needed to propagate a signal is called the minimum Nyquist bandwidth or frequency
• fb = 2B, where fb is the bit rate in bps and B is the ideal Nyquist bandwidth.
• A standard telephone circuit with a bandwidth of 2700 Hz can propagate 5400 bps through it
• Multilevel signaling, where more than two levels are used for signaling allows bit rates exceeding 2B
• Nyquist Formulation: fb = B log2 M
• Rearranged for minimum bandwidth: B= fb/(log2 M ). Bandwidth to pass M-ary digitally modulated carriers
• Simplified, B = fb/N, Baud = fb/N equating baud and ideal bandwidth having the same value and equaling the bit rate divided by bits encoded

Amplitude-Shift Keying (ASK)

• The simplest digital modulation which modulates amplitude of analog carrier directly using a binary information signal
• It is similar to standard amplitude modulation, but has two possible output amplitudes so it is sometimes called digital amplitude modulation (DAM)
• Mod wave: vask(t)= [1+vm(t)]A/2 cos(wt)
• The modulating signal [vm(t)] is a normalized binary waveform, where + 1 V = logic 1 and -1 V = logic 0.
• Therefore, for a logic 1 input, Amplitude = Acos(ωct) and for a logic 0 input, Amplitude = [0]
• Results in carrier being either "on" or "off" yielding the term on-off keying (OOK)
• For every change in the input binary data stream, there is one change in the ASK waveform, and the time of one bit (tb) equals the time of one analog signaling element (t,).
• The rate of change of the ASK waveform (baud) equals the rate of change of the binary input (bps).
• Equation for bandwidth: B = fb /1 = fb

Frequency-Shift Keying (FSK)

• A constant-amplitude form of angle modulation which is similar to standard FM except a binary signal (rather than continuously-changing signal) modulates
• Is sometimes called binary FSK (BFSK) and has the general expression vfsk(t) = Vc cos{2π[fc + vm(t) ∆f]t}
• The peak shift in the carrier frequency (∆f) is proportional to the amplitude of the binary input signal (vm[t]), and the direction of the shift is determined by the polarity.
• Carrier center frequency (fc) is shifted up and down in the frequency domain by the binary input signal
• The output frequency shifts between a mark (logic 1 frequency, fm) and a space (logic 0 frequency, fs)
• Mark and space frequencies are from carrier frequency by the peak frequency deviation (∆f) and from each other by 2 ∆f.
• Use frequency deviation formula: ∆f = |fm – fs| / 2
• The bit time equals the FSK signaling element time, so bit and baud rates equate
• Substitute N = 1 in 2.11: baud = fb / 1 = fb
• Minimum Bandwidth is B = |(fs – fb) – (fm – fb)| = |(fs– fm)| = 2∆f, which can be approximated as B = 2(∆f + fb)

FSK Modulators

• Peak frequency deviation in FSK is constant with highest fundamental being equivalent to half incoming bits, so fa = fb/2 Modulation index for FSK i.e. h-factor = h = Af/fa.
• h = | fm - fs | / fb is another h-factor, with f being the bit rate
• The binary FSK modulator is very similar to conventional FM modulators
• A voltage-controlled oscillator (VCO) is very often used- FSK
• modulator can be operated in the sweep mode where peak frequency deviation = binary voltage & VCO deviator: Af = vm(t) kl, kl being the kilohertz deviation

FSK Receivers

• Noncoherent FSK demodulation is simple, and the FSK input signal is applied to bandpass filters (BPFs) through a power splitter
• Each filter permits mark/space frequency on its’ respective envelope detector and the comparator then acts on the largest magnitude signal
• Coherent FSK receivers can utilize the incoming FSK signal multiplied by a recovered carrier, though not so continuous and not practical
• Phase Locked Loops (PLL) are the most common demodulators: as input shifts between space/mark frequencies, DC error voltage also creates phase shift
• Binary FSK is seldom used for digital radio because performance trails that of PSK or QAM; the use is on analog, voice-band telephone lines

Continuous-Phase Frequency-Shift Keying

• CP-FSK is binary with the mark & space frequencies are synchronized with the input binary bit rate
• Mark/Space are selected so the the separation from center frequency is a multiple of half bit rate; this ensures smooth phase transition
• The conventional or non-continuous form has frequency shifts and abrupt changes yielding less performance
• CP-FSK needs less signal-to-noise in comparison to binary FSK yet requires synchronizing circuits making a more expensive implementation

Binary Phase-Shift Keying

• PSK consists of angle-modulated, constant-amplitude digital modulation
• The simplest is “binary” i.e. BPSK. M=2 thus N=1
• With BPSK, two phases are the carrier so a logic 1 and logic 0 are produced
• The phase of the output carrier shifts between two stages that are 180° separated
• Phase reversal keying or bi-phase modulation are also terms for BPSK; this modulation is a square-wave variant with a CW (continuous wave) output

BPSK Transmitters

• Modulators have input and output in phase
• The balanced modulator reverses phases, it is the switch
• Carrier is transferred to output either in phase or 180° out of phase
• Must operate properly, with a digital input being much being much stronger than the peak value of carrier

BPSK Bandwidth Considerations

• Carriers are changed by binary or modulated
• +/- V input with input of input carrier
• This is represented by +1 and -1 terms, being phase references
• BPSK will shift carrier’s phase between 0°and 180°, thus signaling is equal to information
• To show this BPSK: B= fs, formula, being sin(wc), which will yield lowest double-sided, Nyquist’s at 2fa

BPSK Receivers

• Input may yield +/- sin out output
• Recovery circuit coherent with phase and frequency, with original output
• Modulator is a “product”, with two inputs with “balanced signal”—recovery
• “Low Pass Iteration (LPI)/complex demo-duality is performed, as is recovery, with binary

Quaternary Phase-Shift Keying

• Encoding where N=2, and M=4
• QPSK modulates then will produce input four distinct combinations e.g. 00, 01, 10, 11
• Combines binary input data to form two-bit dibits, yielding +135°, -135°, -45°, +45°

QPSK Transmitters and Bandwidth

• Two bits clock into splitter to produce in-phase signals
• “I”—(inphase)/”Q”—Quadrature bits
• Signals yield 90° out of phase with signal
• Input is given by I to output Balanced

Offset QPSK

• Offset is modified for QPSK, bit waveforms and are offset for single bits
• Midpoints are located at IQ channels
• Conventional change in output means there’ll be corresponding shift in out thus limited to modulation rate

Increasing Efficiency

• bits are encoded to form three bits producing out. To encode, group three incoming bits from 2^3 = 8
• 8PSKmodulator gives bits to IQ conversion
• Bits from "C" channel, and IQ conversion for IQ carriers
• Converters that are 4-level yield higher signals from 2 output levels, digital signal I,Q,C yields converters
• Given in Figure 2.25, the modulator requires one 8PSK
• With data being divided between three, bit rate is 1QC, etc
• In operation, greatest speed happens with 1/4 binary transitions, with "b"
• "O" will equal (sinwI)( sin wC)

16 QAM and Summary

• System where M=16 of four incoming bits from 24-16
• As data is in group-4 with 8QAM, phase, amplitude and transmit are varied
• 16QAM divides the input, but divides 1Q in bit rate = Equal distribution
• For 8PSK, efficiency yields to what is
• C sin(wO)( sin w1)

Band Efficiency

• Bandwidth (Sometimes called Information Density or Spectral). often to compare digital modulation transmission bit rate (bases) Efficiency = minimum bandwidth (Hz) Hertz Where B -Band-width efficiencies
• for data operating determines
• B= 2K BPS per B

Differential PSK

• Information that is conveyed lies in the difference within two signaling channels opposed to actual phase
• DBPSK will generate recovered signal through balanced modulator as does timing
• There is a need to recover bits with reference with “error phase”—Differential Encoding

Error and Rate

• “e” will represent data performance and rate is “bad” when is empirical e.g. when system has bit performance (1.0)

Mathematical Derivations of Eb to No

• P(e) from system data from the performance rating which is known to system (I) and yields ratio
• Noisy power provides "1" in the Hertz range with is related in power density
• E yields “to” No and is what to compare, with modulation and encoding.

Formulas

• C(dBm is power 10 (log to (C))) – No- power/ B" bandwidth is important
• D* = Boltzmann to proportionality 1.4...
• T*= “temperature” at Kelvin
• Noise yields performance for "QAM"
• L *=levels yield axis as "e" for function

Key Points

• Multiplexing in relation with what and can take
• D yields state-region or “point/distance”
• PSK will yield relation for performance by state-change