Fueled by the increased interest in mobile communication for fast moving platforms   , signal detection over fast-fading channels has become an important research area in the last decade   . When signal fading is slow, the channel over at least one symbol interval can be assumed to be Additive White Gaussian Noise (AWGN), and a matched filter receiver front-end followed by symbol rate sampling provides good performance  . However, with fast fading the above matched filter method is suboptimal and more advanced techniques are needed   .
Several methods of receiver design for fast-fading channels have been proposed  -  . Pilot symbol assisted modulation  adds known symbols in the transmitted signal, allowing the receiver to estimate the channel in order to establish an amplitude and phase reference for detection. This technique improves performance; however it lowers the effective bit rate, introduces delay, and requires buffer space at the receiver for channel interpolation. In  it is demonstrated that with fast fading, using a low-pass rectangular pilot filter produces an error floor, and more judiciously designed pilot filters are needed. In  , the authors show that processing more than one sample per symbol ensures robust performance in a fast-fading environment when Nyquist pulse shaping is used, at the expense of increased system complexity compared to traditional detection techniques. In line with such concept a receiver structure for a fading channel applying multisampling is derived in  .
Receivers for fast-fading channels based on filter banks are presented in  -  . In  , the authors demonstrate two types of receivers based on single-filter and double- filter. The single-filter receiver consists of two matched filters derived using a time-se- lective channel model which approximates the fading process by the first two terms of its Taylor expansion. The double-filter receiver consists of two matched filters and two modified matched filters derived using a time-selective channel model approximating the fading process by truncating the Taylor series to the third term. In  , the authors use specific basis functions as receiver filters for discretization. It is claimed that, by a moderate increase in complexity compared to a matched filter receiver, the performance is close to optimal except at very high Signal-to-Noise Ratio (SNR). Another method of designing front-end filters is presented in  , that employs the Karhunen-Loeve (KL) expansion  to approximate the autocorrelation function of the fading process by a finite dimensional separable kernel.
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  that is well suited for fast-fading channels. Furthermore, the Fast Haar Transform (FHT) algorithm  is used to reduce complexity. Performance analysis and Monte-Carlo simulation results are presented for three binary modulation schemes: Time-Orthogonal modulation, Minimum Shift Keying (MSK) and Orthogonal Frequency Shift Keying (FSK).
2. System Model and Discrete Representation of Signals over Time-Varying Rayleigh Channels
2.1. System Model and Framework for Discrete Representation
In this work, we consider a frequency-flat time-varying Rayleigh fading channel, with the complex baseband received signal expressed as 
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  that is based on the KL expansion  . We start with the discretization of the message process of mean
and. The KL expansion for is
where yk are uncorrelated complex Gaussian variables, and the basis functions are obtained by solving the integral equation
In (6) we have
From the properties of the KL representation, we have
where is the Kronecker delta function, and
2.2. Examples for Specific Cases
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
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.
3. Receiver Structure
For convenience, we use the normalized time, expressing the received signal as
where, , and. The symbol denotes quantities in the normalized time setting. For consistency,
must have a single-sided PSD of. The block diagram of the receiver, illustrated in
Figure 1, consists of two parts: a receiver front-end performing the received signal discretization, with output used in the second part that is a MAP detector. In Figure 1 FHT stands for Fast Haar Transform  , and the operator yields a column vector obtained by concatenating the columns of a matrix.
3.1. Receiver Front-End
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  , 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
Figure 1. Receiver block diagram.
where and. In matrix form, (31) can be written as
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  , we have
with, we have
In (38), are normalization factors  , and are a family of unnormalized Haar wavelets  with scaling function
where, , , and mother function 
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  .
3.2. MAP Detector
The observable vector (31) is zero mean jointly Gaussian with conditional Probability Density Function (PDF)
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
3.3. Structural Analysis of the Receiver over Slow-Fading Channels
In this case the fading process satisfies where,
with eigenvalues 
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
Furthermore, since, (4) in the normalized time setting can be expressed as
and substituting this into (7) yields
In (  , 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 (  , p. 170) are in these components. Using (61), the quadratic form in Figure 2 can be written as
Compared to (  , p. 170], our receiver needs to satisfy the following two conditions to achieve optimality:
where is a constant and is defined as
where is a constant matrix, has the form
Figure 2. Simplified receiver block diagram.
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.
4. Performance Analysis for Binary Modulation
4.1. Error Probability
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 
where and are the eigenvalues of. Assuming is transmitted but is detected and denoting, the Pairwise Error Pro- bability (PEP) is 
where is the PDF of. Using similar methods as in 
aided by the residue theorem  , we have from (78)
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 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.
4.2. 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  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  . 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 
The Time-Orthogonal modulation scheme  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  , 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  , 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  in our receiver front-end.
Figure 3. Time-orthogonal modulation, N = 4 and L = 16.
We can find analytically the diversity order that can be obtained with such a Time Orthogonal scheme by using Proposition 2 of  . 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
Figure 4. Positive eigenvalues, N = 4, K = 4, L = 16 and fdT = 0.1.
Figure 5. Negative eigenvalues, N = 4, K = 4, L = 16 and fdT = 0.1.
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  , 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  for MSK modulation. In order to fairly compare our scheme with  , we use (82), which is the same as (  , (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  . Figure 7 illustrates the com- puted and simulated results for MSK with. Comparing with Figure 4 from  , we see that the single-filter receiver yields the same performance as our receiver with K = 2. Comparing with Figure 5 of  , 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  . Our receiver provides better performance since we approximate the fading process more accurately than in  .
Figure 6. MSK modulation (Jakes), N = 4 and L = 64.
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.
Figure 7. MSK modulation (Butterworth), N = 4, L = 64.
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. Moreover, not only the Doppler spread but also the fading spectrum shape affects performance. With Time- Orthogonal modulation, our receiver can achieve optimal performance presented in  as a reference. For MSK, our receiver using four basis functions can lower the error
Figure 8. Orthogonal FSK modulation, N = 4 and L = 64.
floor by more than one order of magnitude compared to the double-filter receiver of  . Orthogonal FSK, which performs the same as Time-Orthogonal modulation over slow fading channels, provides a lower performance over fast time-varying fading channels.
A. Derivation of the Normalization Factors and the Covariance of
We derive the factors that normalize the basis functions to unit energy. We also derive the covariance of defined in (8), using the normalized time setting. From (26) we have
Since with and with, we have
where (96) is obtained using (33), and (97) is due to, being constant over each integration sub-interval with and defined in (43).
The covariance of can be expressed as
Due to and , we have
where (101) is obtained using (33), and (102) is due to, being constant over each integration sub-interval, with and defined in (43).
B. Conditions 1 and 2
We show that satisfying (71) in Section 3.3 is sufficient for Conditions 1 and 2 to hold. Assume that. Because of (71), from (98), we obtain
In this case, from (48) the covariance matrix can be expressed as
and hence, using (67),
where is a constant. Thus, Condition 1 holds.
The inverse of is
where is a constant matrix. Therefore, due to (68) and (109), we have
where is a constant matrix because is a constant matrix. Because of (60), can be expressed as
Due to (67) and (71), (115) becomes
where the non-zero component of can be put in the form
When is constant we have where is a constant. Therefore, Condition 2 also holds.
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