73-78


Al-Khwarizmi Engineering Journal,Vol. 1

 

A Comparison Between 
Recursive Least-Squares (E

Variation

Department of Information and Communication

Email: 

 
(Received 

 
Abstract 
 

In order to select the optimal tracking of fast time var
this paper focuses on the recursive least
reaches the conclusion that E-RLS is more feasible according to the
tracking performance and mean square error 
one time (send/receive) reach to 100 times
 
Keywords: RLS algorithm, E-RLS algorithm, Rayleigh fading channels
 
 
1. Introduction 

 
Data transmission over mobile communication 

channels suffers from fading. The amplitude of 
transmitted signal will changes randomly 
according to affect of Frequency flat fading
important to track variations of the amplitude to 
detect transmitted symbols optimally
nonexistence of the direct line of 
made Rayleigh fading supposition, i.e., the 
amplitude (received amplitude) 
sequence with a Rayleigh density. The fading rate 
of the received signal amplitude is 
by the Doppler bandwidth, which in turn is 
to the speed of mobile or receiver
transmission frequency. This fading rate 
treated through design of powerful 
Subsequently, in the fast single or multipath 
fading channels, cannot be neglect affected part
the amplitude of the signal over a symbol period
The sampling frequency of the received signal 
over slow fading channels is the Nyquist 
frequency of the transmitted signal
sampling rate is twice transmitted signal 
bandwidth, that means the effects of 
varying  Rayleigh fading channel be huge on the 

Khwarizmi Engineering Journal,Vol. 12, No. 1, P.P. 73- 78 (201

 
Comparison Between Recursive Least-Squares (RLS) and Extended

quares (E-RLS) for Tracking Multiple Fast Time 
Variation Rayleigh Fading Channel 

 
Ali Salah Mahdi 

and Communication Engineering/ Al-Khawarizmi College of Engineering
University of Baghdad 

Email: ali_info27@kecbu.uobaghdad.edu.iq 

(Received 25 May 2015; accepted 11 October 2015) 
 

 

In order to select the optimal tracking of fast time variation of multipath fast time variation Rayleigh fading channel
recursive least-squares (RLS) and Extended recursive least-square

is more feasible according to the comparison output of the 
mean square error over five fast time variation of Rayleigh fading

reach to 100 times to make sure from efficiency of these algorithms. 

RLS algorithm, Rayleigh fading channels. 

communication 
The amplitude of 

signal will changes randomly 
of Frequency flat fading. It is 

the amplitude to 
detect transmitted symbols optimally. In the 

direct line of sight, is often 
, i.e., the faded 

 is a random 
The fading rate 
is administered 

in turn is related 
or receiver and the 

fading rate must be 
powerful receivers. 

single or multipath 
not be neglect affected part of 

over a symbol period. 
the received signal 

is the Nyquist 
signal and this 

transmitted signal 
that means the effects of the fast time 

be huge on the 

(faded) signal since the  
bandwidth greater than the 
bandwidth [1,2,3,4], so it 
efficient channel estimation to get the original 
data, here we are compared the performance of 
two algorithms, recursive least
extended recursive least
algorithms to show which one has a good 
to track fast time variation of Rayleigh fading 
channels, to decrease the 
faded signal at the receiver
recent years, the comparison 
tracking performances for
LMS and recursive least-squ
much have been described such as in
LMS algorithm shows
performance than the RLS algorithm because the 
LMS algorithm is model independent, w
RLS algorithm is model dependent.

The remainder of this
follows. We introduced 
Section 2. In Section 3, We present
model of RLS and E-RLS algorithms
Section 4, we compared 
RLS and E-RLS by computer simulation. Section 
5 concludes the paper. 

  
Al-Khwarizmi 
Engineering   

Journal 
(2016) 

quares (RLS) and Extended 
Multiple Fast Time 

Khawarizmi College of Engineering/ 

 

fast time variation Rayleigh fading channel, 
squares (E-RLS) algorithms and 

the simulation program from 
Rayleigh fading channels and more than 

 

  received signal  have a 
than the transmitted data 

[1,2,3,4], so it must be there an 
estimation to get the original 

ere we are compared the performance of 
two algorithms, recursive least-squares (RLS) and 

recursive least-squares (E-RLS) 
to show which one has a good ability 

time variation of Rayleigh fading 
the negative effect on the 

at the receiver. By pointing to the 
the comparison  assessment of the 

for the least mean square 
squares RLS algorithms 

been described such as in [5]–[8]. The 
shows a well tracking 

the RLS algorithm because the 
LMS algorithm is model independent, while the 
RLS algorithm is model dependent. 

The remainder of this paper is prepared as 
 the channel model In 

, We present mathematical 
RLS algorithms. Next, in 

 the performance of the 
RLS by computer simulation. Section 



Ali Salah Mahdi                                Al-Khwarizmi Engineering Journal, Vol. 12, No. 1, P.P. 73- 78 (2016) 
 

74 
 

2. Channel Model 
 

The impulse response of a multipath fading 
channel can be describe as 
 
ℎ��� = ∑ ��	�
� �����	��� − ���                 …(1) 
 
Where 	�,	�� , ����� , are respectively the channel 
length, path loss and Rayleigh fading sequence of 
the �-th reflector groups, and �� are the groups 
delay. In this event, for each	�, assumed the 
amplitude |����| have a Rayleigh distribution as 
shown below [9]: 
 
�|�����	|�|�����	|� =
|�����	|	��|�����	|�/�,					|�����	| ≥	0             …(2) 
 
Whereas, the phase ( )x n∠ is uniformly 
distributed (assumed) with in�−�, �	 : 
 

� ! ( )kx n∠ " = ��# 						, −� ≤ ( )kx n∠ ≤ �   
…(3) 
 
Also, the auto-correlation function of the ����� 
sequence, now considered as a random process, is 
modeled as a zeroth-order Bessel function of the 
first kind [9], i.e., 
 

%��� ≜ '	�����	���� − �(� =
ℐ(�2��+,-�(�,			�( = ⋯ , −1,0,1, …               …(4) 
 
Where ,- is the time sampling of the 
sequence	�����, �+ is the maximum Doppler 
frequency of the Rayleigh fading channel, and the 
function	ℐ(�·� is determined by : 
       

ℐ(�3� ≜ �# 4 cos 	�3 sin :�;:
#
<                         …(5) 

 
 The Doppler frequency �+ is the product of 
carrier frequency �= by speed of mobile users 
(receiver),>, over to speed of light as given below:  
                      

�+ = ?@A=                                                           …(6) 
 

Where  B = 3 ∗ 10E F G⁄ . 
Here, figure (1) show sequence magnitude and 
phase of Rayleigh fading channel at �+ =
100	IJ		, ,- = 1K10�LG, and number of channel 
samples = 200 samples. 

 
 
Fig .1. Sequence magnitude and phase of Rayleigh 
channel  at MN = OPP	QR . 

 
 

3. Mathematical  Model for Tracking Fast 
Time Variation of  Rayleigh fading 
Channel 

 
Here, we have explained the mathematical 

model of two algorithms as below [9]: 
 
A. Tracking fast time variation of Rayleigh fading 
channel via RLS algorithm as given in equations 
below: 
 

ST = ST�� + V
WX⍺Z[WX\[∗

�]VWX\[Z[WX\[∗
�;�^� − _TST�� 		…(7) 

 

T̀ = a�� b T̀�� − V
WXZ[WX\[∗\[Z[WX
�]VWX\[Z[WX\[∗

c                  …(8) 
 
B . Tracking fast time variation of Rayleigh fading 
channel via E-RLS algorithm as given in 
equations below : 
 

ST = ⍺ST�� + V
WX⍺Z[WX\[∗

�]VWX\[Z[WX\[∗
�;�^� − _TST�� …(9) 

 

T̀ = a��|⍺|� b T̀�� − V
WXZ[WX\[∗\[Z[WX
�]VWX\[Z[WX\[∗

c + de		 …(10) 
 

Where 
ST : Weight vector. 
⍺ : Some scalar value	|⍺| ≤ 1. 
a: Some scalar value1 ≪ a ≤ 1. 



Ali Salah Mahdi                                Al-Khwarizmi Engineering Journal, Vol. 12, No. 1, P.P. 73- 78 (2016) 
 

75 
 

T̀: Covariance matrix. 
_T: Input vector 
d: A constant multiple of the identity matrix	e, 
where is more supposed to be less than unit 
magnitude In order to ensure the stability of the 
model. 
 

And the measurements 	;�^� satisfy   
      
	;�^� = _TSTg + >�^�                                    …(11) 
 

ST]�g = ⍺STg + �T                                         …(12) 
 

Where  
STg: Unknown weight vector. 
>�^�: White noise measurement with unit 
variance. 
�T: Disturbance.  
 
 
4. Simulation Result 

 
In this simulation program, we consider the 

channels is multipath Rayleigh fading and fast 
time variation with length 5 taps (number of 
simulated channels), Doppler frequency	�+ =
100Hz, sampling frequency 1.25MHz, number of 
samples for each channel equal to 1000 samples. 
Before displaying the results of simulation, we 
first describe the simulated algorithms parameters 
in Table (1). 
 
Table 1, 
Simulation Parameters. 
 

Parameters Specifications 
All initial values  zero 
Scalar value a 0.995 
Scalar value ⍺  0.95 
Constant value d 0.1 
Number of 
experiments  

100 

 
 

Figure (2), it is explained the magnitude for the 
1st  tap among five channels, it is illustrate three 
lines as described in top left of each figure. 
Furthermore the difference between performance 
of RLS and  E-RLS algorithms to track fast time 
variation of the Rayleigh channel from start to end 
clearly appeared in figure (2), that means,  E-RLS 
algorithm has  a good ability to track fast variation 
than RLS algorithm.  

 
 

Fig. 2. Tracking performance of RLS and E-RLS 
algorithms for Rayleigh channel 1. 

 
 

As well as, we explained tracking performance 
of each algorithm for all other channels as shown 
in Figure (3), Figure (4), Figure (5), And Figure 
(6), still E-RLS superiorly on RLS algorithm. 
 

 
 

Fig. 3. Tracking performance of RLS and E-RLS 
algorithms for Rayleigh channel 2. 
 
 
 
 
 
 
 
 
 
 
 



Ali Salah Mahdi                                Al-Khwarizmi Engineering Journal, Vol. 12, No. 1, P.P. 73- 78 (2016) 
 

76 
 

 
 

Fig. 4. Tracking performance of RLS and E-RLS 
algorithms for Rayleigh channel 3. 

 

 
 

Fig. 5. Tracking performance of RLS and E-RLS 
algorithms for Rayleigh channel 4. 
 

 
 

Fig. 6. Tracking performance of RLS and E-RLS 
algorithms for Rayleigh channel 5. 

Table 2 shows comparison between the mean 
square error MSE values (selected randomly)  for 
the recursive least-squares (RLS) and extended 
recursive least-squares (E-RLS) algorithms and 
the difference it is clearly appeared, MSE of the 
E-RLS (Blue Line) nearest to zero level than MSE 
of the RLS (Black Line). 

 
Table 2, 
The Comparison between Mean square error 
(MSE) of  RLS values and E-RLS values. 
 

MSE 
Sample 
Number 

MSE-
RLS 
Value 

MSE-   
E-RLS 
Value 

Smaller 
Value 

 20 6.1479 3.5628 E-RLS 
200 15.0826 4.3558 E-RLS 
500 16.5332 4.0328 E-RLS 
800 16.9514 2.6490 E-RLS 
1000 17.2023 3.3763 E-RLS 
  
 

Figure (7) described mean square error (MSE) 
for each algorithm. The blue line represent MSE 
of  E-RLS algorithm, and it is nearest to zero level 
than red line (MSE of RLS). Naturally after the 
results that appeared in Figure (3,4), the error 
ratio for     E-RLS should be less than the error 
ratio of RLS algorithm, as shown in the mention 
figure. 

 

 
 

Fig. 7.Mean square error (MSE) of  RLS and E-
RLS algorithms. 

 
 

5. Conclusion 
 

In this paper, we have compared two Blind 
algorithms recursive least-squares (RLS) and 
Extended recursive least-squares (E-RLS), 



Ali Salah Mahdi                                Al-Khwarizmi Engineering Journal, Vol. 12, No. 1, P.P. 73- 78 (2016) 
 

77 
 

adaptive  algorithms mean that, these algorithms 
try to estimate  the channel without need any 
knowledge at the receiver (Blindly). Output 
Figures (2,3,4,5,6) in the simulation section have 
shown a good performance and higher ability for 
E-RLS than RLS to track fast time variation of  
Rayleigh fading channel since the E-RLS blue 
line more close to the original channel red line 
than RLS black line, as well table 2 Explain E-
RLS superiority by comparing output value for 
each algorithms also the simulation program have 
been plotted in the Figure (7). The future work for 
this paper, apply these algorithms to track other 
channels, i.e, finite impulse response complex 
(FIR) channel or Rician fading channels to see 
which algorithm has a good ability to track fast 
amplitude variation than other. 

 
 
6. References 

 
[1] Jayesh H. Kotecha, Petar M. Djuric, "Blind 

sequential detection for Rayleigh fading 
channels using hybrid Monte Carlo-recursive 
identification algorithms", Signal Processing, 
84(5):825-832, 2004.  

[2] J.H. Lodge, M.L. Moher, "Maximum 
likelihood sequence estimation of CPM 
signals transmitted over Rayleigh flat-fading 
channels", IEEE Trans. Commun. 38 (6) 
(June 1990) 787–794. 

[3] G.M. Vitetta, D.P. Taylor, "Multisampling 
receivers for uncoded and coded PSK signal 
sequences transmitted over Rayleigh 
frequency-flat fading channels", IEEE Trans. 
Commun. 44 (2) (February 1996) 130–133. 

[4] H. Zamiri-Jafrian, S. Pasupathy, "Adaptive 
MLSD receiver with identification of flat 
fading channels", International Conference 
on Acoustics, Speech and Signal Processing, 
1997. 

[5] E. Eleftheriou and D. D. Falconer, "Tracking 
properties and steady-state performance of 
RLS adaptive filter algorithms", IEEE Trans. 
Acoust., Speech, Signal Processing, vol. 
ASSP-34, pp. 1097–1110, 1986. 

[6] A. Benveniste, "Design of adaptive 
algorithms for the tracking of time varying 
systems", Int. J. Adaptive Contr. Signal 
Processing, vol. 1, pp.3–29, 1987. 

[7] E. Eweda, "Comparison of RLS, LMS, and 
sign algorithms for tracking randomly time-
varying channels", IEEE Trans. Signal 
Processing, vol.42, pp. 2937–2944, 1994. 

[8] O. Macchi, "Optimization of adaptive 
identification for time-varying filters", IEEE 
Trans. Automat. Contr., vol. AC-31, pp. 283–
287, 1986. 

[9] Ali H. Sayed, "Adaptive Filter", A John 
Wiley & SONS,INC.,PUBLICATIO N, 
2008.    

 
 
 

 
  

 
 

 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

 
 




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