Modulation Technique

Digital Modulation Basics Figures of Merit: Power efficiency . Spectrum / BW efficiency. Complexity. Robustness against impairment, such as : - Linear distortion - Non-linear distortion - Interference - Channel: Fading, Doppler, delay-spread

Digital Modulation Basics Dimension N = 2, in-phase (I) and quadrature (Q) signalling schemes for M-PSK, QAM.

Spectral Efficiency system bit rate / system BW ← b / s / Hz. Where ，Tb : bit duration, Ts : Symbol duration = (log2 M) Tb Power Efficiency: Required signal power Pb (or equivalently, ) for a desired BER. Classical communications may be viewed as a trade-off between Bandwidth & Power Efficiency.

Spectral Efficiency Example 1: To increase η, use M-ary modulation schemes (higher M), since spectral efficiency increases with log2M. But M-ary modulation requires more for same BER as compared with (vis-à-vis) lower M ⇒ Lower Power Efficiency. ∴ Use higher modulation schemes when BW limited (and not power limited). For example voice-band channels.

Spectral Efficiency Example 2: To increase power efficiency, use an Error Correcting Code that will reduce required for required BER; But error correction ⇒ the use of parity check bits ⇒ increased BW or equivalently decreased BW efficiency. ∴ Use error control coding when power limited (and not BW limited). For example satellite channels.

Mobile radio There are two key constraints in choosing modulation schemes for mobile radio. Must be both power and bandwidth efficient. Robust to non-linear distortion. Power amplifiers used for radio transmission tend to be operated in the saturated mode for power efficiency → non-linear distortion.→ Choose constant envelope signalling: FM, PSK, FSK, MSK, ….

Mobile radio ——Optimum receiver for the AWGN channel Modulator maps digital information into signal waveforms within symbol interval of duration T, 0 ≤ t ≤ T.

Mobile radio ——Optimum receiver for the AWGN channel can be represented as a weighted linear combination of orthonormal basis function . N: Dimension of signal set.

Mobile radio ——Optimum receiver for the AWGN channel Fig. Correlator Demodulator

Mobile radio ——Optimum receiver for the AWGN channel Correlator outputs.

Mobile radio ——Optimum receiver for the AWGN channel Correlator outputs.

Mobile radio ——Optimum receiver for the AWGN channel Correlator outputs. ∴ nk → Zero mean Gaussian r.v. and uncorrelated → independent. Variance :

Mobile radio ——Optimum receiver for the AWGN channel Example: M-ary baseband PAM signal set. Basic pulse shape g(t) Signal set has dimension N = 1 only.

Mobile radio ——Optimum receiver for the AWGN channel The basis function f(t) can be formed as Using a correlator type of demodulator, output is:

Mobile radio ——Optimum receiver for the AWGN channel 2d is the distance between adjacent signal amplitudes.

Mobile radio ——Optimum receiver for the AWGN channel Example: M-ary carrier PAM (or ASK) signal set.

Mobile radio ——Optimum receiver for the AWGN channel Using a correlator type of demodulator, output is:

Mobile radio ——Optimum receiver for the AWGN channel The Euclidean distance between any pair of signal points is The distance between a pair of adjacent signal points, i.e., the Minimum Euclidean Distance is

Mobile radio ——Optimum receiver for the AWGN channel Now

Mobile radio ——Optimum receiver for the AWGN channel

Mobile radio ——Optimum receiver for the AWGN channel Placement of thresholds at mid-points of successive amplitude levels.Consider M=4.

Mobile radio ——Optimum receiver for the AWGN channel Given that the m-th symbol is transmitted, the demodulator output is: Let P be the probability of n exceeding in magnitude one-half the distance between levels, that is: Let

Mobile radio ——Optimum receiver for the AWGN channel

Mobile radio ——Optimum receiver for the AWGN channel However, when either of the outside levels ±(M − 1) is transmitted, an error occurs in one direction only.

Mobile radio ——Optimum receiver for the AWGN channel Example 1: Binary PSK - Antipodal signal set - Dimension N = 1 orthonormal basis function f(t)

Mobile radio ——Optimum receiver for the AWGN channel

Mobile radio ——Optimum receiver for the AWGN channel Probability of bit error of PSK

Mobile radio ——Optimum receiver for the AWGN channel

Mobile radio ——Optimum receiver for the AWGN channel Example 2: QPSK

Mobile radio ——Optimum receiver for the AWGN channel

Mobile radio ——Optimum receiver for the AWGN channel Dimension N = 2 → orthonormal basis functions f1(t), f2(t)

Mobile radio ——Optimum receiver for the AWGN channel

Mobile radio ——Optimum receiver for the AWGN channel Probability of bit error of QPSK

Mobile radio ——Optimum receiver for the AWGN channel ∴ QPSK has the same “bit error rate” as BPSK.

Mobile radio ——Optimum receiver for the AWGN channel But QPSK signal has symbol transitions once every Ts = 2 Tb seconds.

Mobile radio ——Minimum Shift Keying (MSK) View MSK as a Continuous-Phase-Frequency-Shift-Keying (CPFSK).

Mobile radio ——Minimum Shift Keying (MSK) is phase angle at t = 0, assumed to be zero. The baseband data b(t) can be written as is bipolar data being transmitted at a rate 1 / T. The tone spacing in MSK is one-half that employed in conventional orthogonal FSK, giving rise to the name “minimum” shift keying.

Mobile radio ——Minimum Shift Keying (MSK) Conventional Orthogonal FSK. Provided, For MSK,

Mobile radio ——Minimum Shift Keying (MSK) Fig. Plot of b ( t ) and θ(t ) for MSK.

Mobile radio ——Minimum Shift Keying (MSK) In each time interval, the phase θ(t ) is piece-wise linear function with slope of either or depending on whether uk is +1 or -1. Here is the modulation index of the MSK. If we extend the linear function to the left, then it will intercept the t =0 axis at a phase value of xk . We therefore can write,

Mobile radio ——Minimum Shift Keying (MSK) This is a piecewise linear phase function of the MSK waveform in excess of the carrier term’s linearly increasing phase (2πfc t ) . xk is a phase constant valid for , determined by the requirement that the phase of the waveform be continuous at the transition instants kT.

Mobile radio ——Minimum Shift Keying (MSK)

Mobile radio ——Minimum Shift Keying (MSK) In above Figure, we plot the linear excess phase function, ，that is valid for for which the data bit is uk −1 and the linear excess phase function, that is valid for that for for which the data bit is uk. At t = kT these two phase functions must attain the same value. That is

Mobile radio ——Minimum Shift Keying (MSK) which leads to a recursive phase constant Set x0 = 0 without loss of generality. ⇒ xk = (0 or π) modulo 2 π

Mobile radio ——Minimum Shift Keying (MSK) xk is the phase axis intercept and is the slope of the linear phase functions over each T second interval. Let u0 = + 1 and u1 = − 1. Then

Mobile radio ——Minimum Shift Keying (MSK) The phase function can be written as where

Mobile radio ——Minimum Shift Keying (MSK) Peak-to-peak frequency deviation divided by bit rate = 0.5.

Mobile radio ——Minimum Shift Keying (MSK) For 0 < t < T, the valid phase function is , where xo is assumed to be 0. For T < t < 2T, the valid phase function is

Mobile radio ——Minimum Shift Keying (MSK) For 2T < t < 3T, the valid phase function is This is the same recurrence relation for xk derived earlier.

Mobile radio ——Minimum Shift Keying (MSK)

Mobile radio ——Minimum Shift Keying (MSK)

Mobile radio ——Minimum Shift Keying (MSK) Over each T second interval, the phase of the MSK waveform is advanced or retarded precisely 90o with respect to carrier phase, depending upon whether the data for that interval is +1 or -1 respectively. •Since xk is 0 or π modulo 2 π

Mobile radio ——Minimum Shift Keying (MSK) regardless whether uk = +1 or -1.

Mobile radio ——Minimum Shift Keying (MSK) Where Thus, we can view MSK as being composed of two quadrature data channels: In-phase or I-channel ⇒

Mobile radio ——Minimum Shift Keying (MSK) Quadrature or Q-channel ⇒ Now

Mobile radio ——Minimum Shift Keying (MSK)

Mobile radio ——Minimum Shift Keying (MSK) Note that:

Mobile radio ——Minimum Shift Keying (MSK) Therefore Even though the data, uk , can change sign every T seconds, for k even, (k-1)odd, since the data term cos(xk ) cannot change sign for k going from odd to even, that is at the zero crossing of S(t).

Mobile radio ——Minimum Shift Keying (MSK) For k odd, (k-1) even, cosxk can only change sign for k going from even to odd, that is at the zero crossing of C(t), provided uk = − uk−1 . Since the data term cannot change value at the zero crossing of C(t) (k even to odd).

Mobile radio ——Minimum Shift Keying (MSK) Finally, for k even, (k-1) odd, That is the data term can only change value at the zero crossing of S(t) (k odd to even). Thus the symbol weighting in either I or the Q channel is a half cycle sinusoidal pulse of duration 2T seconds and alternating sign. The I and Q channel pulses are skewed T seconds with respect to one another.

Mobile radio ——Minimum Shift Keying (MSK) The data are conveyed at a rate of 1/ 2T bps in each of the I and Q channel by weighting the I and Q channel pulses by cos(xk) and ukcos(xk), respectively. Recall since xk = 0 or π modulo 2π, cos(xk) and ukcos (xk) take on only the values ±1.

Mobile radio ——Minimum Shift Keying (MSK)

Mobile radio ——Minimum Shift Keying (MSK)

Mobile radio ——Minimum Shift Keying (MSK) Bit Error Probability of MSK

Mobile radio ——Minimum Shift Keying (MSK)

Mobile radio ——Minimum Shift Keying (MSK)

Mobile radio ——Minimum Shift Keying (MSK)

Mobile radio ——Minimum Shift Keying (MSK)

Mobile radio ——Minimum Shift Keying (MSK) Although GSM uses coherent demodulation, it is a very complex process. In a commercial FM system, usually a simple FM discriminator demodulates the signal. On the other hand, the generation of an MSK signal using the VCO, and the need to control the peak-to-peak frequency deviation to be exactly 0.5 times the bit rate is not an easy process. Using a programmable read only memory(PROM) or a high speed DSP, the generation of an MSK waveform in their quadrature modulation format, I(t)cosωct and Q(t)sinωct is very easy.

Mobile radio ——Minimum Shift Keying (MSK) Therefore an ideal combination would be as shown in the following Figure. The binary information uk is first encoded into cos(xk) and ukcos(xk) . Multiplying these with C(t) and S(t) , respectively, gives I(t) and Q(t). Then forming I(t)cosωct and Q(t)sinωct and summing them to get the MSK waveform, SMSK(t) .

Mobile radio ——Minimum Shift Keying (MSK) Demodulation by a simple FM discriminator produces the original uk . An FM discriminator produces at its output, the derivative of the phase of the signal at its input. If where , then output is

Mobile radio ——Minimum Shift Keying (MSK)

Gaussian-Minimum-Shift-Keying (GMSK)

Gaussian-Minimum-Shift-Keying (GMSK)

Gaussian-Minimum-Shift-Keying (GMSK) Invented by Murota and Hirade1 of NTT, Japan. Binary data are filtered by a Gaussian lowpass filter before they are used to frequency modulate a voltage controlled oscillator (VCO). Aim is to reduce bandwidth of modulated signal, yGMSK(t). Premodulation lowpass filter has a transfer function Bo is the 3-dB bandwidth of the lowpass filter.

Gaussian-Minimum-Shift-Keying (GMSK) at f = Bo , The ratio The impulse response of the lowpass filter can be obtained by taking the inverse Fourier Transform of H(f).

Gaussian-Minimum-Shift-Keying (GMSK) Now let a rectangular pulse p(t) be defined as having a unit amplitude in the time interval, , and zero everywhere. The output of this Gaussian filter with this rectangular pulse at its input is,

Derivation of GMSK pulse equation The rectangular pulse is given as The impulse response of the Gaussian filter is

Derivation of GMSK pulse equation Then the pulse equation of GMSK is

Derivation of GMSK pulse equation Let , then Therefore where

Derivation of GMSK pulse equation Since hence g(t) can be expressed as, for all t.

Derivation of GMSK pulse equation

Derivation of GMSK pulse equation

Derivation of GMSK pulse equation

Derivation of GMSK pulse equation

Derivation of GMSK pulse equation Premodulation LPF should have the properties: 1. Narrow bandwidth and sharp cut-off → to suppress high frequency components. 2. Low overshoot impulse response → to avoid excessive instantaneous frequency deviation. 3. Preservation of the output pulse area which corresponds to a phase shift of π / 2 → for coherent demodulation to be applicable, as simple MSK.

Derivation of GMSK pulse equation Fig. Instantaneous frequemcy variation of GMSK

Derivation of GMSK pulse equation

Derivation of GMSK pulse equation Fig. Measured power spectra of GMSK (V:10dB/div, H:10kHz/div)

Derivation of GMSK pulse equation

Derivation of GMSK pulse equation Fig. Power spectra of GMSK

Derivation of GMSK pulse equation Fig. Fractional power ratio of GMSK

π/4 DQPSK This modulation is used in the American and Japanese digital cellular standards. π/4 DQPSK signal can be expressed as:

π/4 DQPSK φk’s are absolute phase of the carrier signal and Δφk’s are the Gray encoded differential phases.

π/4 DQPSK Example: Assume φ0=0, what would be φk , Ik andQk, when bit stream 00101111…. are sent using π/4 DQPSK?

π/4 DQPSK

DQPSK DQPSK signal can be expressed as:

DQPSK φk’s are absolute phase of the carrier signal and Δφk’s are the Gray encoded differential phases.

DQPSK Example: Assume φ0=0, what would be φk , Ik andQk, when bit stream 00101111…. are sent using DQPSK?

DQPSK Note that DQPSK 4 / π has eight possible transmitted phases whereas DQPSK has only four possible transmitted phases.

DQPSK Advantages of / 4 π-DQPSK over DQPSK The transitions in the signal constellation do not pass through the origin. The envelope exhibits less variation than DQPSK ⇒ better output spectral characteristics. Power amplifier operated in saturated mode for high power efficiency will cause spectral broadening when signal envelope has large fluctuation as in DQPSK

DQPSK Fig. Quaternary DPSK carrier waveform and its delayed version

DQPSK Fig. Comparison of Spectra of Narrowband Modulations

Error probability of Digital Modulation in slow flat fading channels Flat fading → Multiplicative (gain) variation in the transmitted signal envelope. Consider binary BPSK, and assume receiver can obtain an accurate estimate of δ(t), therefore for coherent demodulation, (that is the receiver can compensate for or remove δ(t)), we can consider r(t) to be of the form

Error probability of Digital Modulation in slow flat fading channels Using a matched filter receiver

Error probability of Digital Modulation in slow flat fading channels Slow fading implies channel characteristic changes at a much slower rate than the applied modulation, or the data rate → δ(t), α(t) constant over at least one symbol duration. Matched filter or correlator output r = s + n

Error probability of Digital Modulation in slow flat fading channels First we evaluate error probability of this BPSK modulation conditioned on a particular α.

Error probability of Digital Modulation in slow flat fading channels Now α is Rayleigh distributed Let Signal-to-Noise Ratio (SNR) = What is the pdf of γ?

Error probability of Digital Modulation in slow flat fading channels

Error probability of Digital Modulation in slow flat fading channels ∴ BER of BPSK in slow Rayleigh fading

Differential PSK (DPSK) DBPSK (M = 2) DQPSK (M = 4)

Differential PSK (DPSK)

Differential PSK (DPSK) Demodulation

Differential PSK (DPSK)

Differential PSK (DPSK) We can write Form

Differential PSK (DPSK) In the absence of noise, the phase difference yields the transmitted information → The evaluation of the error probability performance of DPSK is extremely difficult. Very difficult to get the pdf for the phase ψ, of the random variable For M = 2 → BDPSK

Differential PSK (DPSK) BPSK DBPSK

Bit Error Rate

Fig. Probability of error for several systems in Raleigh fading