Uplink Channel Sounding in TDD-OFDM Cellular Systems

Abstract

In this paper, we consider uplink channel sounding in time division duplex (TDD) orthogonal frequency division multiplexing (OFDM)-based cellular systems. The channel information provided by uplink sounding may be necessary to exploit closed loop transmission techniques in TDD-OFDM systems. However, conventional sounding schemes may suffer from large sounding overhead, significantly restricting the number of sounding users due to the resource shortage. To reduce the sounding overhead, the proposed scheme transmits sounding signal only through subchannels which are not highly correlated to each other. The whole channel information can be estimated from the partial one by exploiting the channel correlation in the frequency domain. Thus, the proposed scheme allows a larger number of users to transmit the sounding signal, while making the base station employ closed loop transmission techniques with the use of whole channel information. Finally, simulation results show that the proposed scheme noticeably improves the performance over conventional schemes in the presence of channel correlation.

Appendices

Appendix 1: Derivation of (14)

Assuming that $X$ and $\hat{{X}}$ are zero mean complex Gaussian random variables, $X$ can be modeled with $\hat{{X}}$ and another zero mean complex Gaussian random noise $Z$ with unit variance as [23]

$$\begin{aligned} X=\alpha \hat{{X}}+\beta Z \end{aligned}$$

(45)

where $\alpha$ and $\beta$ are respectively a complex coefficient and a real coefficient, satisfying

$$\begin{aligned} E\left\{ {X\hat{{X}}^{*}}\right\}&= \alpha E \left\{ {\left| {\hat{{X}}}\right| ^{2}}\right\} ,\end{aligned}$$

(46)

$$\begin{aligned} E\left\{ {\left| X\right| ^{2}}\right\}&= \left| \alpha \right| ^{2}E \left\{ {\left| {\hat{{X}}}\right| ^{2}}\right\} + \beta ^{2}. \end{aligned}$$

(47)

It can be seen from (5) and (9) that $\tilde{H}_k (f,m)$ is a complex Gaussian random variable since it is a linear combination of complex Gaussian random variables [23]. Thus, $H_k (f,m)$ can be expressed by $\tilde{H}_k (f,m)$ with the use of (45) where $X=H_k (f,m)$ and $\hat{{X}}=\tilde{H}_k (f,m)$. It can be seen from (9) that

$$\begin{aligned} E\left\{ {H_k (f,m)\tilde{H}_k^*(f,m)}\right\}&= E\left\{ {\tilde{H}_k (f,m)\tilde{\mathbf{H}}_{k,\mathrm{LS}}^*(\mathbf{g}_{k,f},m)}\right\} \left\{ {\mathbf{A}_k^{-1} (\mathbf{g}_{k,f})}\right\} ^{*}\mathbf{C}_k^*(f,\mathbf{g}_{k,f})\nonumber \\&= \mathbf{C}_k(f,\mathbf{g}_{k,f})\left\{ {\mathbf{A}_k^{-1} (\mathbf{g}_{k,f})}\right\} ^{*}\mathbf{C}_k^*(f,\mathbf{g}_{k,f}), \end{aligned}$$

(48)

$$\begin{aligned} E\left\{ {\left| {\tilde{H}_k (f,m)}\right| ^{2}}\right\}&= \mathbf{C}_k (f,\mathbf{g}_{k,f})\mathbf{A}_k^{-1} (\mathbf{g}_{k,f})E \left\{ {{\tilde{\mathbf{H}}}_{k,\mathrm{LS}} (\mathbf{g}_{k,f}, m){\tilde{\mathbf{H}}}_{k,\mathrm{LS}}^*(\mathbf{g}_{k,f}, m)}\right\} \nonumber \\&\quad \times \left\{ {\mathbf{A}_k^{-1} (\mathbf{g}_{k,f})}\right\} ^{*}\mathbf{C}_k^*(f,\mathbf{g}_{k,f}) \nonumber \\&= \mathbf{C}_k (f,\mathbf{g}_{k,f}) \left\{ {\mathbf{A}_k^{-1} (\mathbf{g}_{k,f})}\right\} ^{*}\mathbf{C}_k^*(f,\mathbf{g}_{k,f}). \end{aligned}$$

(49)

Then, $\sigma _k^2 \left( f\right)$ can be represented as

$$\begin{aligned} \sigma _k^2 \left( f\right)&= E\left\{ {\left| {\tilde{H}_k (f,m)-H_k (f,m)}\right| ^{2}}\right\} \nonumber \\&= E\left\{ {\left| {H_k(f,m)}\right| ^{2}}\right\} +E\left\{ {\left| {\tilde{H}_k (f,m)}\right| ^{2}}\right\} -E\left\{ {\left| {\tilde{H}_k (f,m)H_k^*(f,m)}\right| ^{2}}\right\} \nonumber \\&\quad -E\left\{ {\left| {H_k (f,m) \tilde{H}_k^*(f,m)}\right| ^{2}}\right\} \nonumber \\&= 1-\mathbf{C}_k (f,\mathbf{g}_{k,f})\mathbf{A}_k^{-1} (\mathbf{g}_{k,f}) \mathbf{C}_k^*(f,\mathbf{g}_{k,f}). \end{aligned}$$

(50)

Thus, (14) can be derived from (45) where

$$\begin{aligned} \alpha&= \frac{E\left\{ {H_k (f,m)\tilde{H}_k^*(f,m)}\right\} }{E\left\{ {\left| {\tilde{H}_k (f,m)}\right| ^{2}}\right\} }=1,\end{aligned}$$

(51)

$$\begin{aligned} \beta&= \sqrt{E\left\{ {\left| {H_k(f,m)}\right| ^{2}} \right\} -\left| \alpha \right| ^{2}E\left\{ {\left| {\tilde{H}_k(f,m)} \right| ^{2}}\right\} } \nonumber \\&= \sqrt{1-\left( {1-\sigma _k^2 \left( f\right) }\right) } =\sigma _k \left( f\right) . \end{aligned}$$

(52)

Appendix 2: Derivation of (19)

Assuming that $\sigma _k^2 (f)\simeq \sigma ^2$, it can be seen from (18) that

$$\begin{aligned} \left\| {{\tilde{\mathbf{H}}}_{\hat{{k}}_f} (f)}\right\| ^{2}&= \mathop {\max }\limits _{k=\left\{ {0,\ldots ,K-1}\right\} } \left\{ {\left\| {{\tilde{\mathbf{H}}}_k (f)}\right\| ^{2}}\right\} \nonumber \\&= \mathop {\max }\limits _{k=\left\{ {0,\ldots ,K-1}\right\} } \left\{ {\sum _{m=0}^{M-1} {\left| {\tilde{H}_k \left( {f,m}\right) } \right| ^{2}}}\right\} \nonumber \\&\le \sum _{m=0}^{M-1} {\mathop {\max }\limits _{k=\left\{ {0,\ldots , K-1}\right\} } \left\{ {\left| {\tilde{H}_k \left( {f,m}\right) }\right| ^{2}}\right\} } \end{aligned}$$

(53)

and that

$$\begin{aligned} \bar{{\gamma }}&= \gamma _0 E \left\{ {\left\| {{\tilde{\mathbf{H}}}_{\hat{{k}}_f} (f)}\right\| ^{2}}\right\} +\gamma _0 \sigma ^2\nonumber \\&\le M\gamma _0 E\left\{ {\mathop {\max }\limits _{k= \left\{ {0,\ldots ,K-1}\right\} }\left\{ {\left| {\tilde{H}_k \left( {f,m}\right) }\right| ^{2}}\right\} }\right\} +\gamma _0 \sigma ^2. \end{aligned}$$

(54)

Let $\lambda _k \left( {f,m}\right) =\big |{\tilde{H}_k (f,m)}\big |^{2}$ and $\hat{{\lambda }}\left( {f,m}\right) = \max _{k=\left\{ {0,\ldots ,K-1}\right\} }\left\{ {\lambda _k \left( {f,m}\right) }\right\}$ where $\tilde{H}_k (f,m)$ is a complex Gaussian random variable with variance $1-\sigma ^{2}$. Then, it can be seen that $\lambda _k \left( {f,m}\right)$ is a Rayleigh random variable [23] and that $\hat{{\lambda }}\left( {f,m}\right)$ has cumulative distribution function and probability density function as, respectively,

$$\begin{aligned} F_{\hat{{\lambda }}\left( {f,m}\right) } \left( \lambda \right)&= \left\{ {F_{\lambda _k \left( {f,m}\right) } \left( \lambda \right) }\right\} ^{K}\nonumber \\&= \left( {1-\exp \left( {-\frac{\lambda }{1-\sigma ^2}}\right) } \right) ^{K},\end{aligned}$$

(55)

$$\begin{aligned} f_{\hat{{\lambda }}\left( {f,m}\right) } \left( \lambda \right)&= \frac{d}{d\lambda }F_{\hat{{\lambda }} \left( {f,m}\right) }\left( \lambda \right) \nonumber \\&= \frac{K}{1-\sigma ^2}\exp \left( {-\frac{\lambda }{1-\sigma ^2}} \right) \left( {1-\exp \left( {-\frac{\lambda }{1-\sigma ^2}}\right) } \right) ^{K-1} \end{aligned}$$

(56)

where $\exp \left( x\right)$ is exponential function. Then,

$$\begin{aligned} E\left\{ {\hat{{\lambda }}\left( {f,m}\right) }\right\}&= \int \limits _0^\infty {\lambda f_{\hat{{\lambda }} \left( {f,m}\right) } \left( \lambda \right) d\lambda } \nonumber \\&= \int \limits _0^\infty K \frac{\lambda }{1-\sigma ^2}\exp \left( {-\frac{\lambda }{1-\sigma ^2}}\right) \left( {1-\exp \left( {-\frac{\lambda }{1-\sigma ^2}}\right) }\right) ^{K-1}d\gamma . \end{aligned}$$

(57)

Assuming that $x=1-\exp \left\{ {-\lambda /{\left( {1-\sigma ^{2}}\right) }}\right\}$, it can be seen that $\lambda =-\left( {1-\sigma ^{2}}\right) \ln \left( {1-x}\right)$ and $dx=\left( {1-\sigma ^{2}}\right) ^{-1}\exp \left\{ {-\lambda /{\left( {1-\sigma ^{2}}\right) }}\right\} d\lambda$. Thus, $E\left\{ {\hat{{\lambda }}\left( {f,m}\right) }\right\}$ can be derived as

$$\begin{aligned} E\left\{ {\hat{{\lambda }}\left( {f,m}\right) }\right\}&= \int \limits _0^1 K \left( {1-\sigma ^2}\right) x^{K-1}\left( {-\ln \left( {1-x}\right) }\right) dx \nonumber \\&= \int \limits _0^1 K \left( {1-\sigma ^2}\right) x^{K-1}\sum _{k=1}^\infty {\frac{x^{k}}{k}} dx\nonumber \\&= \left( {1-\sigma ^2}\right) \sum _{k=1}^\infty {\int \limits _0^1 K \frac{x^{K+k-1}}{k}dx} \nonumber \\&= \left( {1-\sigma ^2}\right) \sum _{k=1}^\infty {\frac{K}{k \left( {k+K}\right) }} \nonumber \\&= \left( {1-\sigma ^2}\right) \sum _{k=1}^\infty {\left( {\frac{1}{k} -\frac{1}{k+K}}\right) } \nonumber \\&= \left( {1-\sigma ^2}\right) \sum _{k=1}^K {\frac{1}{k}}. \end{aligned}$$

(58)

Then, (19) can be derived from (54).

Appendix 3: Comparison of DS and CSS for the Sounding

As illustrated in Fig. 7, sounding signals from multiple users are multiplexed in the same subchannel by means of DS (i.e., frequency division multiplexing) or CSS (i.e., code division multiplexing), and then simultaneously transmitted to the BS [8, 24]. The DS determines a set of sounding subcarriers for user $k, {\hat{\mathbf{f}}}_{k,\mathrm{DS}}$, as (23). Each user transmits distinguishable sounding signal using orthogonal frequency resource with transmission power $P_{k,\mathrm{DS}}$ determined as (24). The channel at a sounding subcarrier, $H_k (\hat{{f}}_k, m)$, can be estimated by means of LS estimation, yielding a MSE of $\hat{{\sigma }}_{k,\mathrm{DS}}^2 (\hat{{f}}_k)$ as (25).

On the other hand, the CSS determines a set of sounding subcarriers for user $k, {\hat{\mathbf{f}}}_{k,\mathrm{CSS}}$, as

$$\begin{aligned} {\hat{\mathbf{f}}}_{k,\mathrm{CSS}} =\left\{ {\hat{{f}}_k \left| {\hat{{f}}_k =\hat{{b}}_k F+{f}^{\prime }, \text{ for }\,\hat{{b}}_k \in {\hat{\mathbf{B}}}_k \text{ and } \text{0 }\le {f}^{\prime }\le F-1}\right. }\right\} \end{aligned}$$

(59)

and transmits distinguishable sounding signal using a unique orthogonal cyclic shift code as

$$\begin{aligned} S_k (\hat{{f}}_k)&= \sqrt{P_{k,\mathrm{CSS}}}s_k \left( f\right) \nonumber \\&= \sqrt{\frac{P_{\mathrm{t}}}{\hat{{B}}_k F}}\exp \left( {-\frac{j2\pi u_k {f}^{\prime }}{\varphi }}\right) \end{aligned}$$

(60)

where $\varphi \,\left( {\le F}\right)$ is the phase shift index [8]. The channel $H_k (\hat{{f}}_k, m)$ where $\hat{{b}}_k F\le \hat{{f}}_k \le \hat{{b}}_k F+F-1$, can be estimated by means of LS estimation as [21]

Fig. 7

Sounding structure of DS and CSS when $D=\varphi =6$ and $F=18$

$$\begin{aligned} \tilde{H}_{k,\mathrm{LS}} (\hat{{f}}_k,m)=\frac{\hat{{B}}_k F}{\varphi P_{\mathrm{t}}}\sum _{{f}^{\prime }=\varphi i}^{\varphi \left( {i+1}\right) -1} {Y(\hat{{b}}_k F+{f}^{\prime },m)S_k^*(\hat{{b}}_k F+{f}^{\prime })} \end{aligned}$$

(61)

where $i=\left\lfloor {{\mathrm{mod}(\hat{{f}}_k,F)}/\varphi }\right\rfloor$, yielding an MSE of [16]

$$\begin{aligned} \hat{{\sigma }}_{k,\mathrm{CSS}}^2 (\hat{{f}}_k)=I_{k,\mathrm{CSS, self}} (\hat{{f}}_k)+I_{k,\mathrm{CSS}} (\hat{{f}}_k)+\frac{\hat{{B}}_k F}{\varphi P_\mathrm{t}}. \end{aligned}$$

(62)

Here $I_{k,\mathrm{CSS, self}} (\hat{{f}}_k)$ is the self-interference term due to the frequency selective fading represented as

$$\begin{aligned} I_{k,\mathrm{CSS},\text{ self }} \left( {\hat{{f}}_k}\right)&= E \left\{ {\left| {H_k\left( {f,m}\right) -\frac{1}{\varphi }\sum _{{f}^{\prime } =\varphi i}^{\varphi \left( {i+1}\right) -1} {H_k \left( {\hat{{b}}_k F+{f}^{\prime },m}\right) }}\right| ^{2}}\right\} \nonumber \\&= 1+\frac{1}{\varphi }-\frac{2}{\varphi }\sum _{{f}^{\prime }=0}^{\varphi -1} {\text{ Re }\left\{ {R_k \left( {\mathrm{mod}\left( {\hat{{f}}_k, \varphi }\right) -{f}^{\prime }}\right) }\right\} }\nonumber \\&\quad +\,\frac{2}{\varphi } \sum _{\Delta f=1}^{\varphi -1} {\left( {1-\frac{\Delta f}{\varphi }} \right) \text{ Re }\left\{ {R_k \left( {\Delta f}\right) }\right\} } \end{aligned}$$

(63)

and $I_{k,\mathrm{CSS}} (\hat{{f}}_k)$ is the multi-user sounding interference due to the code collision represented as

$$\begin{aligned} I_{k,\mathrm{CSS}} (\hat{{f}}_k)&= \frac{1}{\varphi ^{2}}E \left\{ {\left| {\sum _{{k}^{\prime }=0,{k}^{\prime }\ne k}^{K-1} {\sum _{{f}^{\prime } =\varphi i}^{\varphi \left( {i+1}\right) -1} {H_{{k}^{\prime }} (\hat{{b}}_k F+{f}^{\prime },m)} \exp \left( {j\frac{2\pi \left( {u_k -u_{{k}^{\prime }}} \right) {f}^{\prime }}{\varphi }}\right) }}\right| ^{2}}\right\} \nonumber \\&= \frac{K-1}{\varphi }+\frac{2}{\varphi ^{2}}\sum _{{k}^{\prime }=0,{k}^{\prime } \ne k}^{K-1} \sum _{\Delta f=1}^{\varphi -1} \left[ \begin{array}{l} \text{ Re }\left\{ {R_{{k}^{\prime }} \left( {\Delta f}\right) }\right\} \cos \left( \displaystyle {\frac{2\pi \left( {u_k -u_{{k}^{\prime }}}\right) \Delta f}{\varphi }} \right) \\ -\text{ Im }\left\{ {R_{{k}^{\prime }} \left( {\Delta f}\right) }\right\} \sin \left( \displaystyle {\frac{2\pi \left( {u_k -u_{{k}^{\prime }}}\right) \Delta f}{\varphi }}\right) \end{array}\right] .\nonumber \\ \end{aligned}$$

(64)

where $i=\left\lfloor {{\mathrm{mod}(\hat{{f}}_k, F)}/\varphi }\right\rfloor , \text{ Re }\left\{ x\right\}$ and $\text{ Im }\left\{ x\right\}$ are respectively the real and the imaginary part of a complex number $x$. It can be seen that both $I_{k,\mathrm{CSS, self}} (\hat{{f}}_k)$ and $I_{k,\mathrm{CSS}} (\hat{{f}}_k)$ increase as $R_k \left( {\Delta f}\right)$ decreases.

Assuming that $D=\varphi$, it can be shown from (25) and (62) that the difference between the MSE of the DS and the CSS with the use of LS estimation can be represented as

$$\begin{aligned} \Delta \hat{{\sigma }}_k^2 (\hat{{f}}_k)&= \hat{{\sigma }}_{k,\mathrm{CSS}}^2 (\hat{{f}}_k)-\hat{{\sigma }}_{k,\mathrm{DS}}^2(\hat{{f}}_k) \nonumber \\&= I_{k,\mathrm{CSS, self}} (\hat{{f}}_k )+I_{k,\mathrm{CSS}} (\hat{{f}}_k) \ge 0. \end{aligned}$$

(65)

It can be seen from in (11) and (43) that the MSE of the MMSE interpolation decreases as that of the LS estimation at $\hat{{f}}_k$ decreases. Thus, it is desirable to use the DS rather than the CSS when the MMSE interpolation is employed.