Tài liệu Adaptive WCDMA (P5) docx

After the signal despreading, vector of parameters θ to be estimated includes timing of the received symbols τ0, phase of the received carrier θ0, frequency offset of the received signal

Modulation and demodulation

5.1 MAXIMUM LIKELIHOOD ESTIMATION

We start again with the ML principle defined in Section 3.1 of Chapter 3 After the

signal despreading, vector of parameters θ to be estimated includes timing of the received

symbols τ0, phase of the received carrier θ0, frequency offset of the received signal ν0,

amplitude of the signal A0 and data symbols a n

by equation (3.5) now becomes

 ∞

−∞r(t)h(t − nT − εT − ˜τ) dt (5.6)

Copyright ¶ 2003 John Wiley & Sons, Ltd.

ISBN: 0-470-84825-1

then equation (5.5) becomes

In the special, important case of nonstaggered signals (ε = 0), we find q = p If we define

cn = a n + jb n, the correlation integral becomes

5.1.1 Phase and frequency correction: phase rotations and NCOs

For a given phase error θ (n), the complex signal sample (sampling index n) zin(n) is rected by multiplying the sample by a complex correlation factor exp(j θ (n)) as follows:

cor-zin(n) = xin(n) + jyin(n)

By using exp(j θ ) = cos θ + j sin θ, we get

z0 = xincos θ − yinsin θ + j (xinsin θ + yinsin θ ) ( 5.10)

The operation is known as phase rotation and the block diagram for the realization ofequation (5.10) is shown in Figure 5.1

Frequency corrections (translations) can be performed by the same circuitry but now

the phase correction will change in time For the frequency error ν, the correction becomes

+ _

+ +

yin(n)

Sine/

Cosine ROM

A simultaneous phase rotation and frequency translation is performed as

In the next section we will focus on the problem of detecting phase and frequency error.The circuit from Figure 5.1 will be used for error corrections, given an error value ofphase or frequency

In the simple case of binary modulation we have

La,b=

N−1

m=0cosh

Equation (5.22) can be maximized by changing ˜ν in p(m) in the open loop search By

taking the derivative of equation (5.20), we get the tracker for the QPSK signal

where p v (m)∂p(m)/∂ ˜v A sample of frequency-error detector control signal is

uv(n) = Re[p v(n)] tanh

h ( −t )

−j 2pth(−t )

Frequency loop filter

Frequency error detector {pv(n)}, {qv(n)}

uv(n) {p (n)}, {q (n)}

and its time average U v = E n [u v (n)] is called the detector characteristic or S-curve Theblock diagram realizing equation (5.24) is shown in Figure 5.2

5.2.1 QPSK tracking algorithm: practical version

Further simplification is obtained if we use

tanh(x) ∼ = x, |x|  1

∼

which for nonoffset QPSK results in

u v (n) = Re[p(n)] Re[p v (n)]+ Im[p(n)] Im[p v (n)]

= Re[p(n)p∗

5.2.2 Time-domain example: rectangular pulse

After considerable labor, the S-curve defined as u v(n) = E n [U v (n)] for a unit-amplituderectangular pulse in time domain and random data is found to be [1]

If the data pattern is assumed to rotate by 90◦ from one symbol to the next, that is,

c m+1= jc m, the S-curve is as given in Figure 5.4

For binary phase shift keying (BPSK) dotting signal where c m+1= −c m, the S-curve

is as given in Figure 5.5

One should notice that for random data, S-curve demonstrates regular shape with the

slope decreasing with the timing error Even for nonsynchronized systems with τ /T =

0.5, the system would operate For rotating data the impact of timing is larger For dotting

signal, if the timing error becomes large, the S-curve not only has a reduced slope, which

is equivalent to reducing the signal-to-noise ratio in the loop, but also changes the signresulting in the devastating effects of generating the control signal that would cause loss

0.125

0.25 0.5

5.3 CARRIER PHASE MEASUREMENT:

NONOFFSET SIGNALS

In this case possible solutions will depend very much on a number of parameters.Regarding the signal format, there will be differences for single amplitude [M-ary phase-shift keying (MPSK)] versus multiamplitude [M-ary quadrature amplitude modulation(MQAM)] or offset versus nonoffset signal Different representations such as rectangularversus polar representation of phase error or parallel versus serial representation of signal(offset only) will result in different solutions Additional knowledge such as clock timing(clock-aided) or data or decisions [data-aided (DA) or decision-directed (DD)] will also

be of great importance Configurations such as feedforward (FF) versus feedback (FB)will also offer different advantages and drawbacks

5.3.1 Data-aided (DA) operation

In this case a preamble c n and timing τ0 are available and equation (5.8) becomes

and we define again the sample of the S-curve as

If we represent the output of the phase rotator p(n)e −j ˜θ as the complex sequence

{x(n), y(n)}, the output of the decision algorithm as ˆc n = ˆa n + j ˆb n = sgn(x n) + jsgn(y n)

Loop filter

where sgn(v) = +1 (−1) if v is greater than 0 (v < 0), then equation (5.30) is given by

un(n) = Im
(x n + jy n)( ˆa n + j ˆb n)  = y n sgn(x n) − x n sgn(y n) ( 5.33)

This is known as four-phase hard-limiting Costas detector that is so widely used inQPSK systems

Rectangular representation

If we use the following steps:

• exp(−j ˜θ) → a rectangular representation in equation (5.28)

differentiate with respect to ˜θ

• bring all the expressions into the summation sign

• take the real part of the derivative

• the ML estimate ˆθ occurs for the value of ˜θ at which the derivative goes to zero

circum-signal-to-noise ratio in such systems will be reduced by factor (1–2Pe)2 where Pe is thebit error rate Omitting the decision operation might reduce equipment complexity (Notlikely to be a good reason in a digital implementation Indeed, we find digital DD meth-ods are often simpler than non-DD methods.) In this case the likelihood function will beaveraged out with respect to data

The starting point is equation (5.18) in the form

L( ˜ θ ) = C3exp 2C2

N0Re

( 5.39)

and its derivative can be represented as



2C2

N0Re[e−j ˜θ p(n)]



2C2

N0Re[e−j ˜θ p(n)]

and its implementation is shown in Figure 5.8

For a FF operation we start with

e−j ˆθ = cos ˆθ − j sin ˆθ

tanh x ∼ = x for small x

and equation (5.41) gives

r (t )

Signal in

Phase rotator

cˆn

e −j q~

Solving the last line for ˆθ gives the desired result:

filter

r (t )

Phase rotator

Delay

÷ Table

( ) 2

2

1 2

This result is equivalent to the difference between the outputs of two BPSK Costasdetectors If we use the approximation

where ρ and ψ are the amplitude and the phase of the polar representation of p(n) As

the generalization of equation (5.52) carrier phase is estimated as

Timing

Sample

Data out

r (t )

Phase rotator

Delay Signal in

÷ Table

arctan 1 M

Table

Matched

filter

Decisions p(n ) =

τˆ

cˆn

e −j q~

5.4 PERFORMANCE OF THE FREQUENCY

AND PHASE SYNCHRONIZERS

A uniform representation of the input signal is used by modifying equation (5.14) as follows:



The output of the pulse-matched filter can be represented as



|f | − 12T



1− α 2T < |f | ≤ 1+ α

For the three modulation formats considered in this section, we have

• BPSK : q(m : ˜τ) = 0

• QPSK : q(m : ˜τ) = p(m : ˜τ)

• OQPSK : q(m : ˜τ) = p(m : ˜τ + T /2) = p(m + 1/2 : ˜τ)

The spectra of h(t) is presented in Figure 5.11.

One should be aware that the tracking error variance for the linearized tracking

system is proportional to loop noise to signal power ratio σ2

θ ∝ B LN (f T )/slope2=

BL N (f T )/S Noise power is proportional to noise density and loop bandwidth and nal power to the square of the slope of the equivalent S-curve We will also use notationslope2 = S These parameters are shown in Figures 5.12 to 5.23.

sig-The noise power spectral density of the decision-directed maximum likelihood (DDML)

detector for BPSK and QPSK signals is shown in Figure 5.12 with E s /N0being a eter The same results for OQPSK signal is shown in Figure 5.13

param-The slope of the phase error discriminator S-curve for different modulations is shown

in Figure 5.14

The normalized noise power spectral density for the DDML scheme is shown in Figures5.15 and 5.16 To get the tracking error variance results, from Figure 5.15, should bemultiplied by the loop bandwidth

0.0 0.2 0.5 0.75 1.00

α = 0.5 0.0

QPSK BPSK

f(u) = u RO = 1.0 f(u) = u RO = 0.5 f(u) = 1 RO = 1.0 f(u) = 1 RO = 0.5

f (u) = u and f (u) = 1.

logarithmic scale.

0.0 2.5 5.0 7.5 10.0

f(u) = u RO = 1.0 f(u) = u RO = 0.5 f(u) = 1 RO = 1.0 f(u) = 1 RO = 0.5

0.0 2.5 5.0 7.5 10.0

DDML NDAML V&V, f (u) = u ∗∗2

θ /B L) DDML, NDAML and V&V with

f (u) = u2 for QPSK, logarithmic scale.

Var ∼= (2B L T )A(N o / 2E s ) = A · CRB

carrier synchronization Degradation [dB] from CRB

DDML, NDAMLB/Q, V&VB/Q (same as CRB)

The results for NDAML logarithm are shown in Figures 5.17 and 5.18 The same set

of results for V&V algorithm is shown in Figures 5.19 to 5.22

Comparison of different algorithms is shown in Figure 5.23 and Table 5.1 In the tablethe results are compared to Cramer-Rao Bound (CRB), which is the best achievable result

A number of specific solutions and results related to carrier estimation are given inReferences [3–32]

p (,) – output of pulse-matched filter (I channel)

q (,) – output of pulse-matched filter (Q channel)

a n – data (I channel)

b n – data (Q channel)

z n – complex signal envelope

x , y – real, imaginary part of z

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