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This paper proposes a wavelet based receiver structure for frequency-flat time-varying Rayleigh channels, consisting of a receiver front-end followed by a Maximum A-Posteriori (MAP) detector. Discretization of the received continuous time signal using filter banks is an essential stage in the front-end part, where the Fast Haar Transform (FHT) is used to reduce complexity. Analysis of our receiver over slow-fading channels shows that it is optimal for certain modulation schemes. By comparison with literature, it is shown that over such channels our receiver can achieve optimal performance for Time-Orthogonal modulation. Computed and Monte-Carlo simulated performance results over fast time-varying Rayleigh fading channels show that with Minimum Shift Keying (MSK), our receiver using four basis functions (filters) lowers the error floor by more than one order of magnitude with respect to other techniques of comparable complexity. Orthogonal Frequency Shift Keying (FSK) can achieve the same performance as Time-Orthogonal modulation for the slow-fading case, but suffers some degradation over fast-fading channels where it exhibits an error floor. Compared to MSK, however, Orthogonal FSK provides better performance.

Fueled by the increased interest in mobile communication for fast moving platforms [

Several methods of receiver design for fast-fading channels have been proposed [

Receivers for fast-fading channels based on filter banks are presented in [

In this paper, we present a wavelet based receiver for frequency-flat time-varying Rayleigh channels, consisting of two parts: a front-end stage and a Maximum A-Post- eriori (MAP) detector. Discretization of the received continuous time signal is an essential function of the front-end stage, and for this task we employ the framework for discrete representation of continuous time signals from [

In this work, we consider a frequency-flat time-varying Rayleigh fading channel, with the complex baseband received signal expressed as [

where

where

The process of discretization yields a finite dimensional vector of observables from a segment of a continuous time signal. We use the framework of [

since

because

and

where y_{k} are uncorrelated complex Gaussian variables, and the basis functions

In (6) we have

where

From the properties of the KL representation, we have

where

Slow-Fading Channel with Linear Combination of Orthogonal Signals

The fading process

where

with

where

where

combinations of

Multiplying both sides of (15) by

because of (13). In matrix form, (16) becomes

where

with

We see that (17) is a matrix eigen-problem that can be solved by a multitude of methods.

Orthogonal signaling is a particular case where

Therefore

and the matrix

Substituting (20), (21) and (22) into (15) and using (19) yields

Frequency-Flat Fast-Fading Rayleigh Channel

Consider a basis functions

where the coefficients

Substituting (25) into (7), we have

where

where

where

For convenience, we use the normalized time

where

must have a single-sided PSD of

Operating on

The basis functions

where

where

Using

where

Defining

with

where

with

In (38),

and

where

Next, we define

where

since

where

For conceptual simplicity, we take

where

when R is large, we can use the FHT algorithm that has a computational complexity

The observable vector (31) is zero mean jointly Gaussian with conditional Probability Density Function (PDF)

where

with

The normalization factors ensuring

The structure of the MAP detector can be simplified by using the log-domain

Since

In this case the fading process satisfies

and (35) is of the form

Hence, (36) becomes

Because

and (32) becomes

Therefore, we have

since

zeros, and assuming

we have

Furthermore, since

and substituting this into (7) yields

where

In ( [

Compared to ( [

Condition 1

where

Condition 2

where

and

with

From section B of the Appendix we have that satisfying

where

transmitted signals

and equiprobable

thogonal signaling in the normalized time setting, we have

From (73), it is seen that

1 and 2 hold, showing that our receiver with orthogonal signaling is optimal for slow- fading channels. Next we consider the performance over fast-fading channels.

From (53), using the log-likelihood metrics for hypotheses

Thus, we have the log-likelihood decision rule

Defining

Hermitian quadratic form where

where

where

aided by the residue theorem [

for

for

In our work the PEP was calculated from (79) (80) using the MATLAB software package. We consider two fading autocorrelation functions: the Jakes’ model [

and autocorrelation function of a Butterworth filtered fading process [

where

propable

The covariance matrix ^{−23}. The eigenvalues

where, using (81) and (82), we have

Computer simulations in this paper employ the Monte-Carlo method and are implemented in the C language. We implemented the receiver of

For the Jakes’ model, we use the Rayleigh fading channel simulator of [

sample, the Jakes’ model is expressed as

ples taken per symbol interval. For the Butterworth lowpass filtered fading process, each fading realization is generated by passing two white and independent real Gaussian processes through two identical third-order Butterworth filters as in [

The SNR for simulations can be expressed as

After passing the received signal through an ideal band-limiting anti-aliasing filter, the power spectral density of

with

plex Gaussian with independent real and imaginary components which are stationary with same autocorrelation function, the variance of its real (or imaginary) component is given by [

The Time-Orthogonal modulation scheme [

the MSK modulation scheme can be represented by

and Orthogonal FSK modulation is defined by

All three modulation schemes have the same average energy. According to our observations, we have that

We can find analytically the diversity order that can be obtained with such a Time Orthogonal scheme by using Proposition 2 of [

where for

error floor. Using

From

This paper considers a wavelets based receiver structure for frequency-flat time-varying Rayleigh channels. The receiver consists of a front-end performing discretization of the received continuous time signal, and a MAP detector processing the outputs from the front-end. The fast Haar transform algorithm is used to reduce computational complexity. We present two conditions for achieving optimality over slow-fading channels, and demonstrate that using any orthogonal signaling scheme ensures optimality of our receiver in this case.

Numerical performance analysis and Monte-Carlo simulation results of three binary modulation schemes are presented for fast-fading Rayleigh channels. Among these schemes, Time-Orthogonal modulation performs best, and MSK worst. Increasing K, the number of basis function that the receiver uses, improves performance, but when K > 4 the performance is not improved further for Time-Orthogonal modulation and Orthogonal FSK using the Jakes’ fading model with

floor by more than one order of magnitude compared to the double-filter receiver of [

Shao, X. and Leib, H. (2016) A Receiver Structure for Frequency-Flat Time-Varying Rayleigh Channels and Performance Analysis. Int. J. Communications, Network and System Sciences, 9, 387-412. http://dx.doi.org/10.4236/ijcns.2016.910033

We derive the factors

Since

where (96) is obtained using (33), and (97) is due to

The covariance of

Due to

where (101) is obtained using (33), and (102) is due to

We show that satisfying (71) in Section 3.3 is sufficient for Conditions 1 and 2 to hold. Assume that

In this case, from (48) the covariance matrix can be expressed as

and hence, using (67),

where

The inverse of

where

where

Due to (67) and (71), (115) becomes

where the non-zero component of

When

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