In this work, we consider a frequency-flat time-varying Rayleigh fading channel, with the complex baseband received signal expressed as [18]

where, () is transmitted with a-priori probability, is the fading process and is additive noise. The processes and are zero mean complex Gaussian and mutually independent. We assume that and have independent real and imaginary components that are stationary with same autocorrelation function. We also assume that is white with a single-sided power spectral density (PSD). We can express (1) in the form

where, is a random M-dimensional vector with a-priori probability, and having 1 as the mth component with the others being 0. Essentially, the vector selects the signal that is transmitted, and it is independent of and.

The process of discretization yields a finite dimensional vector of observables from a segment of a continuous time signal. We use the framework of [16] that is based on the KL expansion [15] . We start with the discretization of the message process of mean

since, autocorrelation

because

and. The KL expansion for is

where y_k are uncorrelated complex Gaussian variables, and the basis functions are obtained by solving the integral equation

In (6) we have

where

From the properties of the KL representation, we have

where is the Kronecker delta function, and

Slow-Fading Channel with Linear Combination of Orthogonal Signals

The fading process where g has zero mean and autocorrelation, and is expressed as

where are complex scalars and are orthogonal real functions such that

with denoting the energy of. In this case, (4) can be written as

where. After substituting (14) into (7), we have

where, showing that the basis functions are linear

combinations of. The variables and can be found by solving a matrix eigen-problem.

Multiplying both sides of (15) by and integrating results in

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

where, , ,

with being a block diagonal matrix, and

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 and hence

Therefore

and the matrix in (17) is, showing that

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

Frequency-Flat Fast-Fading Rayleigh Channel

Consider a basis functions for the frequency-flat fast-fading Rayleigh channel. From (4), we see that the kernel can be of infinite dimension because of the auto- correlation function. When approximating as a N dimensional separable kernel, (4) becomes

where the coefficients are calculated to yield a good approximation, and are suitable real functions. Denoting, where, (24) becomes

Substituting (25) into (7), we have

where. We see that the basis functions are now linear combinations of. The coefficients of the linear combinations can be formed by solving a matrix eigen-problem. Multiplying both sides of (26) by and integrating yields

where. In matrix form, (27) becomes

where, , and We see that (28) is also a matrix eigen-problem. After sub- stituting and into (26), we can compute the basis functions.

Operating on, the front-end stage produces the observable vector with com- ponents

The basis functions can be found using the second example in Section 2.2. In the normalized time setting, the parameters and functions in (24) are selected using the wavelet-based eigenfunction method in [16] , and hence and where and are eigenvalues and eigenfunctions of the autocorrelation function. Substituting (26) using the nor- malized time setting into (30), we have

where and. In matrix form, (31) can be written as

where and.

Using with and

where with c the maximum wavelet level, denoting a family of normalized Haar wavelets including the scaling function with corresponding coefficients that can be found using the method from [16] , we have

Defining

with, we have

where

with, and

In (38), are normalization factors [19] , and are a family of unnormalized Haar wavelets [19] with scaling function

and

where, , , and mother function [19]

Next, we define as the resolution of, and hence we can divide (38) into R sub-integrals resulting in

where, () are samples of given by

since are constant over each integration sub-interval. From (42), we have

where and with

For conceptual simplicity, we take since a larger R does not affect the value of. Therefore, from (44), we have

where and To solve (46)

when R is large, we can use the FHT algorithm that has a computational complexity where N is the number of input elements [17] .

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

where and

with where is defined in (8) using the normalized time setting, , and

The normalization factors ensuring have unit energy are derived in the Appendix. In (48), is obtained by using

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

Since is constant, finding the maximum value of is equivalent to finding the minimum over the M log-likelihood metrics

In this case the fading process satisfies where,

with eigenvalues [16]

and (35) is of the form

Hence, (36) becomes

Because with from (54) we have

and (32) becomes

Therefore, we have showing that are linear combinations of. From (42) we have

since and for. We can simplify (57) by removing the

zeros, and assuming. Then, with and

we have

Furthermore, since, (4) in the normalized time setting can be expressed as

and substituting this into (7) yields

where

In ( [20] , p. 170], the authors present optimum receivers for slow-fading channels. From Figure 2, it is seen that in order to prove that our receiver can achieve optimality, we need to focus on two components: the quadratic form and the bias term, since the major differences between our receiver and the optimum receiver from ( [20] , p. 170) are in these components. Using (61), the quadratic form in Figure 2 can be written as

Compared to ( [20] , p. 170], our receiver needs to satisfy the following two conditions to achieve optimality:

Condition 1

where is a constant and is defined as

Condition 2

where is a constant matrix, has the form

and

with a constant.

From section B of the Appendix we have that satisfying

where is a constant sufficient for Conditions 1 and 2 to hold. Assume that the

transmitted signals are orthogonal

and equiprobable. From the first example of Section 2.2 when applied to or-

thogonal signaling in the normalized time setting, we have

From (73), it is seen that is constant. Because of (73) and (74), both Conditions

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 (was transmitted) and (was transmitted), the log-likelihood ratio can be expressed as

Thus, we have the log-likelihood decision rule

Defining and the bias term, then is a

Hermitian quadratic form where is a Hermitian matrix because and are Hermitian. The observable vector is zero mean jointly Gaussian, and the conditional PDF of is given in (47). The characteristic function of the Hermitian quadratic form is given by [21]

where and are the eigenvalues of. Assuming is transmitted but is detected and denoting, the Pairwise Error Pro- bability (PEP) is [18]

where is the PDF of. Using similar methods as in [22]

aided by the residue theorem [23] , we have from (78)

for, and

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 [24]

and autocorrelation function of a Butterworth filtered fading process [12]

where is the normalized Doppler spread. We assume that are equi-

propable. From (79) and (80), it is seen that the error performance is determined by which are the eigenvalues of and.

The covariance matrix is given by (48) with its components obtained from (49). The double integrations in (49) are computed numerically using the MATLAB function quad2d with an absolute accuracy of 10⁻²³. The eigenvalues are computed using the function eig which calculates eigenvalues of a symbolic matrix and ensures accuracy to at least 32 significant decimal digits by default. The SNR for performance analysis is defined as

where, using (81) and (82), we have The accuracy of the performance analysis is confirmed by computer simulations.

Computer simulations in this paper employ the Monte-Carlo method and are implemented in the C language. We implemented the receiver of Figure 1 with three binary modulation schemes: Time-Orthogonal modulation, MSK, and Orthogonal FSK. The Bit Error Rate (BER) is estimated from at least 400 errors. In addition, we run at least 10,000 fading channel realizations to ensure accuracy. To emulate continuous time signals we massively oversample by using 4096 samples per symbol interval.

For the Jakes’ model, we use the Rayleigh fading channel simulator of [25] that is based on the sum-of-sinusoids algorithm, where we employ 50 sinusoids. Since we over-

sample, the Jakes’ model is expressed as where, and with S the total number of sam-

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 [12] . The 3 dB bandwidth of these filters, , is a measure of the fading rate.

The SNR for simulations can be expressed as where

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

with which is the sampling frequency. Instead of sampling at rate, oversampling by a factor S yields. Since the additive noise is zero mean com-

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 [26]

The Time-Orthogonal modulation scheme [13] is defined by the waveforms

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 is large enough for approximating well the fading autocorrelation functions for these cases. Moreover, for Time-Orthogonal modulation and for MSK as well as Orthogonal FSK are large enough to achieve good performance. Thus, in this paper, we use these parameter settings for simulations. Unless explicitly stated, we use the Jakes’ model for performance analysis and simulations.

Figure 3 illustrates the computed and simulated BER for Time-Orthogonal modulation with different values of K and. We see that increasing K can improve performance. For the lower Doppler, increasing K beyond 4 does not improve performance for SNR less than 50 dB. For larger Doppler, using more than four basis functions can slightly improve performance for SNR > 40 dB. In [13] , the authors propose a receiver front-end using specific basis functions to discretize the received continuous time signal, which is simple to implement. In order to show that this receiver has a close to optimal performance, the authors also provide the optimal performance for as reference. Comparing Figure 3 with [13] , we see that our receiver can achieve optimal performance for, 4 and 6. To reduce the overall complexity of our scheme we use the FHT algorithm, whose computational complexity is with N the number of input elements [17] in our receiver front-end.

We can find analytically the diversity order that can be obtained with such a Time Orthogonal scheme by using Proposition 2 of [22] . Essentially the result of this proposition is

where for the parameter is the sum of all positive eigenvalues of, that has one additional eigenvalue at −1 with multiplicity. Figure 4 and Figure 5 present the magnitude of these eigenvalues on a log scale for Time-Orthogonal modulation with and. Figure 4 shows that two positive eigenvalues increase linearly with SNR, and Figure 5 shows that two negative eigenvalues decrease with SNR, converging to −1. In our case, we have two distinct and positive eigenvalues of multiplicity () satisfying, and a negative eigenvalue of −1 at with multiplicity 2. Hence, this scheme provides an asymptotic diversity order of two which correlates well with our results of Figure 3 for.

Figure 6 illustrates the calculated and simulated BER for MSK modulation with different values of K and. For, we see that using two basis functions leads to a high error floor, and increasing K to 4 can improve performance and remove the

error floor. Using can slightly improve performance further for SNR > 60 dB. For, it is seen that using yields a higher error floor compared to the case, and increasing K to 4 can lower the error floor by three orders of magnitude. Using six basis functions can further improve performance and remove the error floor. We see that increasing the normalized Doppler spread degrades performance in this case. In [12] , the authors present single and double-filter receivers designed for linearly and quadratically time-selective Rayleigh fading channel models. These receivers correspond to our case of two and four basis functions respectively. Performance analysis and simulation results are presented in [12] for MSK modulation. In order to fairly compare our scheme with [12] , we use (82), which is the same as ( [12] , (5.1)), to design the basis functions for the receiver. We generate the fading process using two identical third-order Butterworth lowpass filters as in [12] . Figure 7 illustrates the com- puted and simulated results for MSK with. Comparing with Figure 4 from [12] , we see that the single-filter receiver yields the same performance as our receiver with K = 2. Comparing with Figure 5 of [12] , we see that the double-filter receiver provides the same performance as our receiver with K = 4 for SNR < 40 dB. For larger SNR, however, our receiver performs better and has an error floor that is more than one order of magnitude lower than the error floor in [12] . Our receiver provides better performance since we approximate the fading process more accurately than in [12] .

From Figure 6 and Figure 7, we see that, for MSK with, the fading spectrum shape affects the performance of our receiver when using K > 2. With the Jakes’ fading spectrum, there is no error floor for and the improvement between K = 4 and K = 6 is small. With the Butterworth fading spectrum, the performance is worse than with Jakes’ fading, and there exists an error floor for K = 4. When K = 6, we do not observe an error floor for SNR £ 70 dB. With Butterworth fading we see a larger performance improvement when increasing K from 4 to 6 than with Jakes’ fading.

Figure 8 illustrates the computed and simulated BER for Orthogonal FSK. We see that for, using K = 2 leads to a high error floor, while increasing K improves performance and removes the error floor. However, beyond K = 4, increasing K does not improve performance for SNR less than 50 dB. For we see that using K = 2 results in a higher error floor compared to, and increasing K to 4 removes the error floor for SNR below 60 dB. Using K = 6 can improve performance further for SNR > 35 dB. Orthogonal FSK and Time-Orthogonal modulation are orthogonal signaling schemes with same performance over slow fading channels. However when increases, Orthogonal FSK performs worse than Time-Orthogonal modulation, and better than MSK.