SIGNAL DESIGN FOR LS AND MMSE CHANNEL ESTIMATORS
Pertti Järvensivu, Marja Matinmikko, Aarne Mämmelä
VTT Electronics, Kaitoväylä 1, P.O. Box 1100, FIN-90571 Oulu, Finland, Pertti.Jarvensivu@vtt.fi
Abstract - We analyzed the effect of time and frequency
domain windowing, or weighting, of the transmitted signal
for least squares (LS) and minimum mean-square error
(MMSE) estimators when the channel is time-variant. We
considered the estimation error for different windows and
we found that the windows can be selected independently.
We explained the performance of the estimators with the
help of the radar ambiguity function of the transmitted
signal including the windowing.
Keywords - Doppler-delay-spread function, matched filter
bank, square-root raised cosine, Hamming, Blackman.
I. INTRODUCTION
In this paper, we analyze the effect of time and frequency
domain windowing of the transmitted signal on the perfor-
mance of least squares (LS) and minimum mean-square
error (MMSE) estimators. We also present some simulation
results. We use Bello’s [1] estimator of the Doppler-delay-
spread function, which is the Fourier transform of the output
delay-spread function with respect to the time variable [2].
Bello’s work was an extension of Levin’s [3] work in time-
invariant channels to time-variant channels.
Bello’s estimator includes a matched filter bank, a set of
samplers and an equalizer for the sidelobes of the ambiguity
function of the transmitted signal (Fig. 1). The estimator can
be easily extended to estimation of the output delay-spread
function by using a set of inverse discrete Fourier transforms
[4]. The MMSE estimator corresponds to the estimator pres-
ented in [5], which estimates the input delay-spread function
(also known as the time-variant impulse response) instead of
the output delay-spread function. A comprehensive review
of channel and other estimators was presented in [6] where
the relationships between different channel estimators are
clearly shown. An earlier review is in [7] which includes
Levin’s estimator.
Implementation of a matched filter bank with the discrete
Fourier transform (DFT) was considered already in [8]. DFT
can be used if the measurement signal is effectively a
rectangular pulse without any modulation. The use of fast
Fourier transform (FFT) in filter banks is well known in the
radar literature [9]. For example, Yao and Cafarella [10]
used a simplified version of Bello’s [1] estimator and the
DFT. They also used a Hamming window in the time
domain in the receiver. However, no mathematical analysis
was given to consider the effects of windowing. Some
theory on time and frequency domain windowing, or
weighting, of a linearly frequency modulated chirp pulse is
presented in [11]. The organization of the rest of the paper is
as follows. In Section II, we present the system model
including the definition of the Doppler-delay-spread func-
tion. We discuss factors affecting the system performance in
Section III. In Section IV, we present some simulation
results. Finally, we make some conclusions in Section V.
Matched
filter bank
Equalizer
r vˆ
(a)
Matched filter
Matched filter
Matched filter
Equal-
izer
Matched filter bank
Sampling
r
vˆ
(b)
Fig. 1. (a) LS and MMSE estimators of Doppler-delay-
spread function (b) Matched filter bank.
II. SYSTEM MODEL
In this paper, we consider random time-variant channels.
The complex envelope of the received signal can then be
written as [2] (Sections III-B and D)
t n d d V e t z
t n d t h t z t n d t g t z t r
t j
,
, ,
2
(1)
where z(t), 0 < t < T is the transmitted signal, g(t,) is the
input delay-spread function, h(t,) is the output delay-spread
function, V(,) is the Doppler-delay-spread function, is
the Doppler shift, is the delay and n(t) is zero mean
additive white Gaussian noise (AWGN). The transmitted
signal z(t) is formed by first applying a single impulse or a
sequence of impulses to a pulse shaping filter with an
impulse response p(t), and then windowing the result with
w(t) (Fig. 2). Although we consider m-sequences here, also
other pseudorandom sequences could be used.
0-7803-7589-0/02/$17.00 ©2002 IEEE PIMRC 2002