Modulation Technique.

Name: Modulation Technique.
Uploaded: 2017-10-09T06:28:52+00:00
Duration: PTM25S0
Channel: Nathan Gilmore
Description: Modulation Technique.

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.

can be represented as a weighted linear combination of orthonormal basis function . N: Dimension of signal set.

Fig. Correlator Demodulator

Correlator outputs.

Correlator outputs. ∴ nk → Zero mean Gaussian r.v. and uncorrelated → independent. Variance :

Example: M-ary baseband PAM signal set. Basic pulse shape g(t) Signal set has dimension N = 1 only.

The basis function f(t) can be formed as Using a correlator type of demodulator, output is:

2d is the distance between adjacent signal amplitudes.

Example: M-ary carrier PAM (or ASK) signal set.

Using a correlator type of demodulator, output is:

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

Now

Placement of thresholds at mid-points of successive amplitude levels.Consider M=4.

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

However, when either of the outside levels ±(M − 1) is transmitted, an error occurs in one direction only.

Example 1: Binary PSK - Antipodal signal set - Dimension N = 1 orthonormal basis function f(t)

Probability of bit error of PSK

Example 2: QPSK

Dimension N = 2 → orthonormal basis functions f1(t), f2(t)

Probability of bit error of QPSK

∴ QPSK has the same “bit error rate” as BPSK.

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).

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.

Conventional Orthogonal FSK. Provided, For MSK,

Fig. Plot of b ( t ) and θ(t ) for 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,

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.

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

which leads to a recursive phase constant Set x0 = 0 without loss of generality. ⇒ xk = (0 or π) modulo 2 π

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

The phase function can be written as where

Peak-to-peak frequency deviation divided by bit rate = 0.5.

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

For 2T < t < 3T, the valid phase function is This is the same recurrence relation for xk derived earlier.

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

regardless whether uk = +1 or -1.

Where Thus, we can view MSK as being composed of two quadrature data channels: In-phase or I-channel ⇒

Quadrature or Q-channel ⇒ Now

Note that:

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).

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).

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.

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.

Bit Error Probability of 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.

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) .

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

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.

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

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

Then the pulse equation of GMSK is

Let , then Therefore where

Since hence g(t) can be expressed as, for all t.

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.

Fig. Instantaneous frequemcy variation of GMSK

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

Fig. Power spectra of GMSK

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 …. 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 …. 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

Using a matched filter receiver

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

First we evaluate error probability of this BPSK modulation conditioned on a particular α.

Now α is Rayleigh distributed Let Signal-to-Noise Ratio (SNR) = What is the pdf of γ?

∴ BER of BPSK in slow Rayleigh fading

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

Demodulation

We can write Form

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

BPSK DBPSK

Bit Error Rate

Fig. Probability of error for several systems in Raleigh fading

Modulation Technique.

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