Received signal with conventional filters

The OFDM radar exploits orthogonal subcarriers to transmit data. It uses the inverse Fourier transform. An OFDM symbol is expressed as [Reference Hakobyan7]

(1)\begin{equation} \forall t \in [0,T], x(t) = \frac{1}{\sqrt{N}}\sum\limits_{n=0}^{N-1} a_n {\rm e}^{\jmath 2\pi f_n t}, \end{equation}

where f_n is the nth subcarrier among a total sum of N subcarriers carrying a_n, which is the nth complex data symbol. The duration of the OFDM symbol is T with $f_n = {n}/{T}$. The cyclic prefix (CP) is added at the beginning of the symbol to ensure the orthogonality of the subcarriers on the receiving end. The symbol from (1) becomes

(2)\begin{equation} \forall t \in [0,T+T_g], x(t) = \frac{1}{\sqrt{N}}\sum\limits_{n=0}^{N-1} a_n {\rm e}^{\jmath 2\pi f_n (t-T_g)}, \end{equation}

where T_g is the duration of the CP. The OFDM radar transmits a chain of OFDM symbols, and the general expression of the signal is

(3)\begin{equation} \forall t\in {\mathbb{R}},x(t) =\frac{1}{\sqrt{N}}\!\sum\limits_{k=-\infty}^{+\infty}\sum\limits_{n=0}^{N-1}a_{k,n} {\rm e}^{\jmath 2\pi f_n (t-k(T+T_g)-T_g)}. \end{equation}

The transmitted signal is

(4)\begin{equation} s(t) = x(t){\rm e}^{\jmath 2\pi f_c t}, \end{equation}

where f_c is the carrier frequency and $a_{k,l}$ is the complex data symbol of the nth subcarrier and the kth OFDM symbol. The bandwidth used by the radar is $B=\frac{N}{T}$. The signal is assumed to be narrowband, which means that $B\ll f_c$.

A target reflects the signal, and the echo is received by the radar. The radial speed of the target is v and its range is $R(t) = R_0+vt$, where R ₀ is the initial range. The delay of the echo is thus

(5)\begin{equation} \tau(t) = \frac{2(R_0+vt)}{c} = \tau_0+\frac{2v}{c}t, \end{equation}

where c is the speed of light and τ ₀ is the delay of the echo due to the range of the target at the beginning of the duration of the symbol. We assume that $\frac{v}{c}\ll 1$. The received and mixed echo is

(6)\begin{align} y(t) &= A\,{\rm e}^{\jmath\varphi}s(t-\tau(t)){\rm e}^{-\jmath 2\pi f_c t}\notag\\ &= A\,{\rm e}^{\jmath\varphi}x(t-\tau(t)){\rm e}^{-\jmath 2\pi f_c \tau(t)}+w(t), \end{align}

with w(t) a white Gaussian noise and $A{\rm e}^{\jmath\varphi}$ the complex amplitude that takes into account the antenna gains, the RF chain, the path loss, and the radar cross-section of the target. For the sake of simplicity, this amplitude is assumed to be $A{\rm e}^{\jmath\varphi}=1$. In this scenario, it is assumed that the variation of $\tau(t)$ through the interval $t\in \left[k(T+T_g)+Tg,(k+1)(T+T_g)\right]$ is negligible. The delay of the kth OFDM symbol is thus $\tau_k = \tau(k(T+T_g)+Tg)$.

In what follows, we make the assumption that T_g is properly sized, which means $\forall k, 0\leq \tau_k \lt T_g\leq T$. The payload of the kth echo is within the interval $\left[k(T+T_g)+Tg,(k+1)(T+T_g)\right]$. Let t ^ʹ be the time within this interval, where $t = k(T+T_g)+T_g+t^\prime$. The kth echo becomes

(7)\begin{equation} y_k(t^\prime) =\frac{1}{\sqrt{N}} \sum\limits_{n=0}^{N-1}a_{k,n} {\rm e}^{\jmath 2\pi f_n (t^\prime-\tau_k)}{\rm e}^{-\jmath 2\pi f_c\tau_k}+w_k(t^\prime). \end{equation}

The received echo is sampled with a sampling period of $T_s = \frac{T}{N}$. The sampled echo is

(8)\begin{align} y[k,m] &=\frac{1}{\sqrt{N}} \sum_{n=0}^{N-1}a_{k,n} {\rm e}^{\jmath 2\pi \frac{n}{T} (\frac{mT}{N}-\tau_k)}{\rm e}^{-\jmath 2\pi f_c\tau_k}\notag\\ &\quad+w[k,m]. \end{align}

Let $\phi_k =- 2\pi f_c\tau_k$. The echo is OFDM demodulated through a Fourier transform and is expressed in the frequency domain as

(9)\begin{align} Y[k,l] &= \frac{1}{\sqrt{N}} \sum\limits_{m=0}^{N-1} y[k,m]{\rm e}^{-\jmath 2\pi \frac{ml}{N}}\notag\\ &= a_{k,l} {\rm e}^{\jmath\phi} {\rm e}^{-\jmath 2\pi \frac{\tau_k l}{T}}+W[k,l]. \end{align}

The Fourier transform of the Gaussian white noise $w[k,m]$ is $W[k,l]$. The samples of both forms are independent and identically distributed random variables, and their variance is $\sigma_w^2$.

The demodulated base band echo $Y[k,l]$ is analyzed to estimate the range of the target. Different filters can be used to this end, among which are the MF and the ZF.

Matched filter

The MF estimates the spectral density of the echo by multiplying it by $\overline{a}_{k,l}$, the conjugate of the complex symbol $a_{k,l}$. An inverse Fourier transform results in a correlation function with a peak at the delay of the target. The kth range profile is [Reference Sen and Nehorai2]

(10)\begin{align} \chi[i] &= \frac{{\rm e}^{\jmath\phi}}{\sqrt{N}} \sum\limits_{l=0}^{N-1} Y[k,l] \overline{a}_{k,l} {\rm e}^{\jmath 2\pi \frac{li}{N}}\notag \\ &= \frac{{\rm e}^{\jmath\phi}}{\sqrt{N}} \sum\limits_{l=0}^{N-1} |a_{k,l}|^2 {\rm e}^{\jmath \frac{2\pi}{N} \left(i-\frac{N\tau_k}{T}\right) l} \notag\\ & \quad+ \frac{{\rm e}^{\jmath\phi}}{\sqrt{N}} \sum\limits_{l=0}^{N-1} W[k,l] \overline{a}_{k,l} {\rm e}^{\jmath 2\pi \frac{li}{N}}. \end{align}

Assuming that the delay τ_k is such that $i_k=\frac{\tau_k N}{T}$ is integer, the peak is then

(11)\begin{align} \chi[i_k] &= \frac{{\rm e}^{\jmath\phi}}{\sqrt{N}} \sum\limits_{l=0}^{N-1} |a_{k,l}|^2\notag\\ &\quad+ \frac{1}{\sqrt{N}} \sum\limits_{l=0}^{N-1} W[k,l] \overline{a}_{k,l}{\rm e}^{\jmath\phi} {\rm e}^{\jmath 2\pi \frac{li}{N}}. \end{align}

Note that in the off-grid scenario (where i_k is not an integer), a spectrum leakage controlling window such as Chebyshev is needed for (11) to be valid, taking into account the Chebyshev window characteristics.

Knowing that $E\left[W[k,l]\right]=0$, the magnitude of the first moment of the range profile is, for all i

(12)\begin{equation} |E\left[\chi[i]\right]| = \frac{1}{\sqrt{N}} \left| \sum\limits_{l=0}^{N-1} |a_{k,l}|^2 {\rm e}^{\jmath \frac{2\pi}{N} \left(i-\frac{N\tau_k}{T}\right) l} \right|. \end{equation}

The magnitude of the first moment of the main lobe is

(13)\begin{align} |E\left[\chi[i_k]\right]| &= \frac{1}{\sqrt{N}} \sum\limits_{l=0}^{N-1} |a_{k,l}|^2\notag\\ &= \sqrt{N} \sigma_a^2, \end{align}

with $\sigma_a^2$ the variance of $a_{k,l}$. It is assumed that N is large enough to verify $\lim\limits_{N\rightarrow +\infty}\sum\limits_{l=0}^{N-1} |a_{k,l}|^2 = N \sigma_a^2$. However, this first approximation is not tight enough, as shown in [Reference Benmeziane, Baudais, Méric and Cinglant5]. We then need to use the second order.

Since the data samples $a_{k,l}$ and the noise samples $W[k,l]$ are mutually independent, the second moment of the range profile is

(14)\begin{align} E\left[\left|\chi[i]\right|^2\right] = \left|E\left[\chi[i]\right]\right|^2 + \sigma_a^2\sigma_w^2. \end{align}

Thus, the average intensity of the range profile for an MF OFDM radar is

(15)\begin{equation} E\left[\left|\chi[i]\right|^2\right] = \frac{1}{N} \left| \sum\limits_{l=0}^{N-1} |a_{k,l}|^2 {\rm e}^{\jmath \frac{2\pi}{N} \left(i-\frac{N\tau_k}{T}\right) l} \right|^2+\sigma_a^2\sigma_w^2 \end{equation}

and the second order of the main lobe is

(16)\begin{align} E\left[\left|\chi[i_k]\right|^2\right] &= \frac{1}{N} \left| \sum_{l=0}^{N-1} |a_{k,l}|^2\right|^2+\sigma_a^2\sigma_w^2\notag\\ &=\frac{1}{N}\sum_{l=0}^{N-1} |a_{k,l}|^4+\frac{1}{N}\!\sum_{l=0}^{N-1}\!\sum_{\substack{l^{\prime}=0\\l^{\prime}\neq l}}^{N-1} |a_{k,l}|^2|a_{k,l^{\prime}}|^2\notag\\ &\quad+\sigma_a^2\sigma_w^2\notag\\ &=\mu_a^4+(N-1)\sigma_a^4+\sigma_a^2\sigma_w^2, \end{align}

assuming N is large enough such that $\mu^4_a=E[|a_{k,l}|^4]$.

Zero forcing

The ZF estimates the transfer function of the channel through an element-wise division [Reference Sturm, Pancera, Zwick and Wiesbeck3]. An inverse Fourier transform gives the impulse response with a peak at the delay of the target. The kth range profile is

(17)\begin{align} \chi[i] &= \frac{{\rm e}^{\jmath\phi}}{\sqrt{N}} \sum\limits_{l=0}^{N-1} \frac{Y[k,l]}{a_{k,l}} {\rm e}^{\jmath 2\pi \frac{li}{N}}\notag \\ &= \frac{{\rm e}^{\jmath\phi}}{\sqrt{N}} \sum\limits_{l=0}^{N-1} {\rm e}^{\jmath \frac{2\pi}{N} \left(i-\frac{N\tau_k}{T}\right) l}\notag\\ &\quad+ \frac{{\rm e}^{\jmath\phi}}{\sqrt{N}} \sum\limits_{l=0}^{N-1} \frac{W[k,l]}{a_{k,l}}{\rm e}^{\jmath 2\pi \frac{li}{N}}. \end{align}

The magnitude of the first moment of the range profile is

(18)\begin{align} |E\left[\chi[i]\right]|= \frac{1}{\sqrt{N}} \left| \sum\limits_{l=0}^{N-1} {\rm e}^{\jmath \frac{2\pi}{N} \left(i-\frac{N\tau_k}{T}\right) l} \right| = \sqrt{N} \mathbb{1}(i = i_k), \end{align}

where $\mathbb{1}(i = i_k)$ is the indicator function that equals 1 when $i = i_k$ and equals zero otherwise. Again for more accuracy, we need to use the second order. The second moment of the range profile is

(19)\begin{align} E\left[\left|\chi[i]\right|^2\right] &=|E\left[\chi[i]\right]|^2 + \frac{1}{\sqrt{N}} \sum\limits_{l=0}^{N-1} \frac{\sigma_w^2}{|a_{k,l}|^2}\notag\\ &=|E\left[\chi[i]\right]|^2 +\sigma_{1/a}^2\sigma_w^2, \end{align}

where $\sigma_{1/a}^2$ is the average of $\frac{1}{|a_{k,l}|^2}$. Finally, the average intensity of the range profile for a ZF OFDM radar is

(20)\begin{equation} E\left[\left|\chi[i]\right|^2\right] = \frac{1}{N} \left| \sum\limits_{l=0}^{N-1} {\rm e}^{\jmath \frac{2\pi}{N} \left(i-\frac{N\tau_k}{T}\right) l} \right|^2+\sigma_{1/a}^2\sigma_w^2 \end{equation}

and the second moment of the main lobe is

(21)\begin{align} E\left[\left|\chi[i_k]\right|^2\right] = N+\sigma_{1/a}^2\sigma_w^2. \end{align}

Now that the range profiles for OFDM/MF and OFDM/ZF have been described through their average intensity in (15) and (20), and the average intensity of their main lobe in (16) and (21), they can be compared via their PSLR and ISLR.

Peak to side lobe ratio and integrated side lobe ratio

The two metrics chosen for the comparison are PSLR and ISLR. The PSLR is the ratio of the power of the main lobe to the power of the highest side lobe. A bad PSLR means that there is a side lobe so high that is could pass the threshold and trigger a false alarm. The ISLR, on the other hand, is the ratio of the integrated main lobe to the integrated side lobes. A bad ISLR means that the main lobe could be below the threshold causing a miss detection.

Let γ be such as

(22)\begin{equation} \gamma^2 = E\left[\frac{|\chi[i_k]|^2}{\theta \left( \{|\chi[i]|^2\}_{i\neq i_k}\right)}\right], \end{equation}

with $\theta(\{x_i\}) = \max\limits_i x_i$ for PSLR and $\theta(\{x_i\}) = \sum\limits_i x_i$ for ISLR. Using the Jensen’s inequality, the expectation of a ratio is approximate by the ratio of the expectations. The expression (22) becomes

(23)\begin{equation} \gamma^2 = \frac{E\left[|\chi[i_k]|^2\right]}{E\left[\theta \left( \{|\chi[i]|^2\}_{i\neq i_k}\right)\right]}. \end{equation}

Since the sampling rate is the nominal frequency, the main lobe is one sample wide. To evaluate γ ², it remains to calculate the sum in (15) and (20). To this end, let $z[i] = {\rm e}^{\jmath 2\pi \left(\frac{i}{N}-\frac{\tau_k }{T}\right)}$. The ratio for the MF filter is

(24)\begin{equation} \gamma^2_\text{MF} = \frac{\mu_a^4+(N-1)\sigma_a^4+\sigma_w^2\sigma_a^2}{\theta \left(\left\{\frac{1}{N}\left|\sum\limits_{l=0}^{N-1}|a_{k,l}|^2z[i]^l\right|^2+\sigma^2_w\sigma_a^2 \right\}_{i\neq i_k}\right)} \end{equation}

and the ratio for a ZF filter is

(25)\begin{equation} \gamma^2_\text{ZF} = \frac{N+\sigma_w^2\sigma_{1/a}^2}{\theta \left(\left\{\frac{1}{N}\left| \sum\limits_{l=0}^{N-1} z[i]^l\right|^2+\sigma_w^2\sigma_{1/a}^2\right\}_{i\neq i_k} \right)}. \end{equation}

Since $|z[i]|=1$ for all i and by approximating the sum in (24) by its average with respect to $a_{k,l}$, it comes

(26)\begin{align} E_a\left[\left|\sum_{l=0}^{N-1}|a_{k,l}|^2z[i]^l\right|^2\right] & = N\mu_a^4+\sum_{l=0}^{N-1}\sigma_a^2z[i]^l\left(\sum_{\substack{l^{\prime}=0\\l^{\prime}\neq l}}^{N-1}\sigma_a^2z[i]^{-l^{\prime}}\right)\notag\\ & =N\mu_a^4+\sigma_a^4\sum_{l=0}^{N-1}\left(\sum_{l^{\prime}=0}^{N-1}z[i]^{l-l^{\prime}}-1\right)\notag\\ & =N(\mu_a^4-\sigma_a^4)+\sigma_a^4\left|\sum_{l=0}^{N-1}z[i]^l\right|^2 \end{align}

and

(27)\begin{equation} \gamma^2_\text{MF} = \frac{\mu_a^4+(N-1)\sigma_a^4+\sigma_w^2\sigma_a^2}{\theta \left(\left\{\frac{\sigma_a^4}{N}\left|\sum\limits_{l=0}^{N-1}z[i]^l\right|^2 +\mu_a^4-\sigma_a^4+\sigma^2_w\sigma_a^2 \right\}_{i\neq i_k}\right)}. \end{equation}

From this point and on, the θ function is approximated by its linear development, $\theta(x)=\eta x$. Using (20) and (27), the relationship between the ZF and the MF is obtained

(28)\begin{align} \gamma^2_\text{MF}=\frac{1+\dfrac{N\sigma_a^4}{\mu_a^4-\sigma_a^4+\sigma_w^2\sigma_a^2}}{1+\dfrac{N}{\sigma_w^2\sigma_{1/a}^2}}\gamma^2_\text{ZF}, \end{align}

for both the PSLR and the ISLR.

In low noise environments where $\sigma_w^2 \rightarrow 0$, $\gamma_\text{MF}^2 \lt \gamma_\text{ZF}^2$ and, more specifically, this happens when

(29)\begin{align} \sigma_w^2 \lt \frac{\mu_a^4-\sigma_a^4}{\sigma_{1/a}^2-\sigma_a^2}. \end{align}

This can be written as

(30)\begin{align} \text{SNR}=\frac{\sigma_a^2}{\sigma_w^2} \gt \frac{\frac{\sigma_{1/a}^2}{\sigma_a^2}-1}{\frac{\mu_a^4}{\sigma_a^4}-1}. \end{align}

It is proposed that, in these low noise environments or high SNR regimes, ZF provides higher PSLR and ISLR compared to MF.

Extension to MMSE filter

The MMSE filter used here is the one applied to multicarrier spread spectrum systems [Reference Fazel and Kaiser6, § 2.1.5.1]. This filter minimizes the mean squared error between the transmitted symbols and the received ones. Applied to the range profile, it leads to

(31)\begin{align} \chi[i] &=\frac{{\rm e}^{\jmath\phi}}{\sqrt{N}}\sum_{l=0}^{N-1}Y[k,l]\frac{\overline{a}_{k,l}}{|a_{k,l}|^2+\sigma_w^2}{\rm e}^{\jmath2\pi\frac{li}{N}}\notag\\ &= \frac{{\rm e}^{\jmath\phi}}{\sqrt{N}} \sum\limits_{l=0}^{N-1} \frac{|a_{k,l}|^2}{|a_{k,l}|^2+\sigma_w^2} {\rm e}^{\jmath \frac{2\pi}{N} \left(i-\frac{N\tau_k}{T}\right) l} \notag\\ & \quad+ \frac{{\rm e}^{\jmath\phi}}{\sqrt{N}} \sum\limits_{l=0}^{N-1} W[k,l] \frac{\overline{a}_{k,l}}{|a_{k,l}|^2+\sigma_w^2} {\rm e}^{\jmath 2\pi \frac{li}{N}}. \end{align}

The same derivations as the ones used for the MF filter in the section “Matched filter” are now followed. The second order moment of (31) follows the same logic as (14), and it is

(32)\begin{align} E[|\chi[i])|^2]=|E[\chi[i]]|^2+\sigma_w^2\sigma_c^2, \end{align}

with $\sigma_c^2=E\left[\frac{|a_{k,l}|^2}{(|a_{k,l}|^2+\sigma_w^2)^2}\right]$, and

(33)\begin{align} |E[\chi[i]]|^2=\frac{1}{N}\left|\sum_{l=0}^{N-1}\frac{|a_{k,l}|^2}{|a_{k,l}|^2+\sigma_w^2}z[i]^l\right|^2. \end{align}

All that is left is to replace $|a_{k,l}|^2$ in (26) by $\frac{|a_{k,l}|^2}{|a_{k,l}|^2+\sigma_w^2}$, and finally

(34)\begin{align} \gamma^2_\text{MMSE}=\frac{1+\dfrac{N\sigma_{b}^4}{\mu_{b}^4-\sigma_{b}^4+\sigma_w^2\sigma_{c}^2}}{1+\dfrac{N}{\sigma_w^2\sigma_{1/a}^2}}\gamma^2_\text{ZF}, \end{align}

where $\sigma_b^2=E\left[\frac{|a_{k,l}|^2}{|a_{k,l}|^2+\sigma_w^2}\right]$ and $\mu_b^4=E\left[\left(\frac{|a_{k,l}|^2}{|a_{k,l}|^2+\sigma_w^2}\right)^2\right]$. This relationship applies for both PSLR and ISLR. Contrary to the MF and ZF parameters, $\mu_a^4$ or $\sigma_{1/a}^2$, the MMSE parameters $\mu_b^4$, $\sigma_b^2$, and $\sigma_c^2$ depend on the noise level, which make it more difficult to predict trends in medium to high noise environments. When $\sigma_w^2=0$, the MMSE parameters are $\sigma_b^2=\mu_b^4=1$ and $\sigma_c^2=\sigma_{1/a}^2$. Thus, the MMSE equals the ZF in low noise environment.

Both expressions (28) and (34) are summarized in only one expression

(35)\begin{align} \gamma^2=\frac{1+\dfrac{N\alpha}{\beta-\alpha+\sigma_w^2\delta}}{1+\dfrac{N}{\sigma_w^2\sigma_{1/a}^2}}\gamma^2_\text{ZF}, \end{align}

with α, β, and δ given in Table 1 for each MF and MMSE metrics.

Table 1. Metric parameters in (35)

Results and discussion

Simulations are performed to validate the expressions given by (28) and (34). Complex 16-QAM data symbol are carried on N = 1024 orthogonal subcarriers spanning a band of B = 375 MHz. The subcarrier spacing is 360 kHz and the OFDM symbol duration is $T=2.73~\mu$s. The center frequency is $f_{\rm c} = 77$ GHz. The duration of the CP is the eighth of the duration of a symbol, $T_g = 0.34~\mu $s. This duration is enough for the sampling window of the receiver end to include a whole symbol for target ranges under 50 m. The target range is R = 12 m, which causes an integer delay. This allows us to go without the use of a Chebyshev window. The OFDM radar parameters for the simulation are summarized in Table 2.

Table 2. OFDM radar parameters

Range profiles are estimated through MF, ZF, and MMSE using (10), (17), and (31), respectively. Then (22) is used to estimate the simulated $\gamma_\text{PSLR}$ and $\gamma_\text{ISLR}$ for each filter. The MF and MMSE performances obtained by simulation are compared to the MF and MMSE performances obtained with (28) and (34), respectively, and using the ZF performances obtained by simulation.

The normalized range profiles are shown superimposed in Figure 1. The figure shows that, for $\text{SNR}=20$ dB, MF exhibits higher side lobes. Note that since MMSE is close to ZF in high SNR environments, it is not plotted here for clarity sake. These results confirm the proposition that in low noise environments, ZF and MMSE have higher PSLR and ISLR compared to MF. Next, the $\gamma_\text{PSLR}$ and $\gamma_\text{ISLR}$ are estimated for different SNR levels, and the results are compared to the proposed expressions. The SNR is the ratio of the symbol power $\sigma_a^2$ to the noise power $\sigma_w^2$. These results are shown in Figure 2 for PSLR and Figure 3 for ISLR.

Figure 1. Normalized and superimposed MF and ZF range profiles.

Figure 2. PSLR versus input SNR.

Figure 3. ISLR versus input SNR.

The plots show three main trends. In the high end of SNR, where $\text{SNR} \gt 10$ dB, (28) is proven to be accurate, since ZF shows a higher level of PSLR and ISLR, while MF plateaus. MMSE exhibits a behavior similar to that of a ZF filter as well as its performances. The expression (35) also proves true. For lower SNR between −20 and 10 dB, the noise is too high for (28) to apply. Therefore, MF displays a slightly higher level or PSLR and ISLR. MMSE behaves more like an MF and (35) proves less accurate than in higher SNR environments. When the $\text{SNR} \lt -30$ dB, the noise level is too high for detection.

The main reason that could explain this difference in low-noise environments is the side lobes due to the correlation function. Even when the noise level is low, these side lobes remain unchanged, forcing the PSLR and ISLR in OFDM/MF radar to plateau. However, the noise floor of an impulse response depends on the noise only.

A few assumptions or approximations are used to obtain the analytical performances: the asymptotic regime with large N, the Jensen’s inequality for the PSLR and ISLR ratios in (23), the average in (26), and the linear development of θ. All these approximations are tight enough to get simple and efficient analytical expressions, as the simulation results show. Figure 4 shows the PSLR versus the number of subcarriers N in both a low-noise environment ($\text{SNR}=30$ dB) and a high-noise environment ($\text{SNR}=-10$ dB). It shows that the expressions are accurate regardless of N.

Figure 4. PSLR versus N, $\text{SNR}\in\{-10,30\}$ dB.

However, when $\tau(t)\neq\tau_k$ through the payload time of the kth echo, i.e. the variation of $\tau(t)$ during this time interval cannot be neglected, the Doppler shift mismatches the filter by distorting the received signal. The stop-and-go approximation in (8) is not more valid, and new derivations are needed. As shown in Figure 5, MF is more robust to this mismatch than ZF: the performance of MF decreases slowly with the Doppler shift in contrast with ZF and MMSE ones. Due to this difference in behavior, the analytical expression (28) is not valid in high SNR environments and significant Doppler. Note that a relative Doppler shift $\nu T=0.1$ corresponds to a radial speed around 260 km/h, with $\nu = \frac{2v f_c}{c}$. Contrary to MF, the analytical MMSE performance matches the simulation one, and (34) remains valid for all Doppler shift conditions. In lower SNR regimes (see Figure 6), the analytical derivations are valid for ZF and MMSE for all Doppler shifts.

Figure 5. PSLR versus Doppler shift, $\text{SNR}=30$ dB.

Figure 6. PSLR versus Doppler shift, $\text{SNR}=10$ dB.