Chapter 3 Proposed Maximum Doppler Shift Estimation Method
3.2 Proposed Doppler Shift Estimation Method Using Doppler Power Spectrum 23
In this section, the procedure of proposed maximum Doppler shift estimation method is introduced. First, we use pilot tones to estimate the CIR. After an observation interval with K OFDM symbols, the consecutive estimated CIRs are used to compute the ACF. And then, we translate the ACF into the frequency domain. Therefore, the Doppler power spectrum of estimated CIRs is obtained. In order to improve the estimated accuracy of the maximum Doppler shift estimation, we could average some Doppler power spectrum estimated results. Then, using the estimated Doppler power spectrum finds the maximum Doppler shift. Figure 3.4 shows a block diagram of proposed maximum Doppler shift estimation method.
Least Square
Figure 3.4: Block diagram of proposed Doppler shift estimation method
Here, we utilize frequency domain scattered pilot tones [1], [21] of DVB-T systems to estimate channel response. Therefore, let L ≤NP be the maximum predicted normalized length of channel. Then, the NP ≥(L+ equally spaced pilots, 1)
kp
P , are inserted at subcarriers k for p 0≤ ≤p (NP −1). Form Equation (3.9), a least square (LS) estimation of the channel transfer function at the pilot subcarriers can be obtained as respectively. Then, through an NP-point IFFT, the estimated CIR hlavg would be
l ( )
In this thesis, we approximate channel time-variations with a piece-wise linear model during one symbol period Tsym. In Figure 3.5, the solid curve is real or imaginary part of a channel path, and the dashed line is piece-wise linear model. By the assumption above, E h
(
lavg −hn l,)
is minimized at n =(N/2−1) for the l-thFigure 3.5: Piece-wise linear model in one OFDM symbol period
From Equation (3.13) and Equation (3.17), the estimated ACF of the channel impulse response of l-th tap can be expressed as
( )
where s is the difference in OFDM symbol index, Tsym denotes the symbol period, and σΔ2 is the combined reduced variance of ICI plus AWGN. Here, the estimated Doppler power spectrum over an observation interval of K symbols can be written as
l ( )K FT
{
hh(
sym) }
If the Doppler power spectrum is approximated over sufficient many observation intervals, the maximum Doppler shift can be estimated by finding the maximum peak of the estimated Doppler power spectrum series. However, excessively long length of observation would lead to an estimation delay and a degradation of the Doppler shift tracking ability.
From Figure 3.3, we can know that the maximum peak of the Doppler power spectrum locates within a small frequency domain interval. Thus, in order to increase the accuracy of maximum Doppler shift estimation, the zero-padding would be used.
The K-sample estimated ACF sequence (Equation 3.18) is extended to a KFFT-sample sequence by padding with (KFFT −K)zero samples. Therefore, we obtain
( )
( ) (
( ))
{ }
FFT FFT 0 , sym , , 1 sym , 0, 0, , 0
K = ⎡⎢⎣φhh φhh T φhh K− ⋅T ⎤⎥⎦
S " " (3.20)
However, we only need a particular small frequency domain interval for Doppler shift estimation. Thus, we introduce the chirp transform algorithm [23] to provide another alternative choice for Fourier transform.
3.2.1 Chirp Transform Algorithm (CTA)
Unlike the FFT scheme, the CTA scheme [22] can be used to compute any set of equally spaced samples of the Fourier transform, thus the CTA scheme is more flexible than the FFT scheme. From Equation (3.19), we denote that
lK ⎡φhh( )0 ,φhh
(
Tsym)
, ,φhh(
(K 1) Tsym)
⎤Φ = ⎢⎣ " − ⋅ ⎥⎦ (3.21)
as a K-point sequence and S is its Fourier transform. Consider the evaluation of M K
samples of that are equally spaced in angle on the unit circle at frequencies SK 0
wk =w + Δ , (3.22) k w
where w is the start frequency and the frequency increment w0 Δ can be chosen arbitrarily. Figure 3.6 shows the frequency samples for the CTA scheme.
Im
Re ω0 (M− Δ1) ω
Unit circle
Figure 3.6: Frequency samples for chirp transform algorithm
The Fourier transform corresponding to the more general set of frequency samples can
Then, substituting Equation (3.22) into Equation (3.23) can obtain l ( ) 1l ( ) 0
where W defined as
W =e− Δj w. (3.25)
Here, we use the identity
( )2
2 2
1
nk = 2⎡⎢⎣n +k − k−n ⎤⎥⎦ (3.26)
to express Equation (3.24) as
l ( ) ( )
However, we need only compute the output of the system in Figure 3.7 over a finite interval for the evaluation of the Fourier transform samples specified in Equation (3.27).
( )
Figure 3.7: Block diagram of chirp transform algorithm
Since g n is finite duration, only a finite portion of the sequence ( ) W−n2/2 is used
However, in Equation (3.29) h n is noncausal, and for certain real-time ( ) implementation it must be modified to obtain a causal system. Therefore, this modification is easily accomplished by delaying h n by (( ) N −1) to obtain a causal
Since both the chirp demodulation factor at the output and the output signal are also delayed by (N −1) samples, the results of the CTA scheme are
lM( ) ( 1 ,) 0,1, ,( 1)
S k =G n+N − n = " M − . (3.32)
By modifying the system of Figure 3.7, we can obtain a causal system as shown in
Figure 3.8: Block diagram of chirp transform algorithm for causal finite-length impulse response
3.2.2 Averaging and Smoothing Effect of Maximum Doppler Shift Estimation Method
If the observation interval with K symbols is not long enough, the Doppler information does not have a sufficient statistical characteristic on the estimated Doppler power spectrum SlM( )k . Therefore, the proposed maximum Doppler shift estimation is overlapped every K/2 samples to deal with this problem, as shown in Figure 3.9.
As the estimated Doppler power spectrum is accumulated, the variance of the noise and the interference power spectrums can be reduced over whole spectrum range and the estimated maximum Doppler shift would become more accuracy.
q-th Observation Interval
(q-1)-th Observation Interval (q+1)-th Observation Interval
(Autocorrelation)
q-th Estimated Doppler Power Spectrum Average
Figure 3.9: Averaging effect of estimated Doppler power spectrum
If the Doppler shift does not change during Q times of observation interval, the ideal Doppler power spectrum can be approximated by averaging Q times of the estimated Doppler power spectrum. Thus, we obtain
i ( ) 1l( )( ) ( ) estimated maximum Doppler shift can be obtained as
i ( )
The Doppler shift estimation above described provides an instantaneous estimated of Doppler shift over Q times of observation interval. The smoothed Doppler spread values over a long period of time can be calculated as [7], [10]
( )
(
( ))
( ) ( ) ( )Here, the absolute difference Δ between the current estimated and the previous f
smoothed estimate can be written as
Δ is a maximum frequency difference. Note that Δfmax depends on update period.