# Impulse response of Time Varying Channel

I have just started studying LTV channels in wireless communication.

I know that $$y(t) = \int _{-\infty} ^\infty x(t-\tau)h(t,\tau)d\tau$$ Is there any way we can calculate the Impulse response $$h(t,\tau)$$ if we know the input $$x(t)$$ and output $$y(t)$$.

Also, can anyone suggest a source to read about the basics of the LTV-Channels? Thanks in Advance.

In the context of wireless communications, the channel impulse response (CIR) is often estimated indirectly via the time-varying transfer function (TVTF) $$H(t, f)$$, defined by: $$H(t, f) = \mathcal F_\tau [ h(t, \tau)]$$ where $$\mathcal F_\tau [ \cdot ]$$ denotes the Fourier transform with respect to the $$\tau$$ (lag) variable.
For example, the air interface for LTE, E-UTRA, uses OFDMA on the downlink. Because OFDM-based schemes do most of the processing in the frequency domain, the TVTF is a more natural representation of the channel. Values of the TVTF at different time-frequency cells are estimated periodically through the use of reference symbols that are transmitted in every resource block. Interpolation may be used to estimate the values of $$H(t, f)$$ at points in between the test points.
If you are interested in estimating the CIR directly in the time-domain, one technique that is used by CDMA-based systems is called a rake receiver, which uses a parallel bank of correlators to measure the multipath components (MPCs) at various delay values (i.e. values of $$\tau$$). When measuring the CIR in the time-domain, the following lowpass channel model is usually assumed: $$h(t, \tau) = \sum_{k=1}^K \alpha_k(t) e^{j \phi_k(t)} \delta( \tau - \tau_k)$$ where $$\alpha_k$$, $$\phi_k$$ and $$\tau_k$$ represent the amplitude shift, phase shift, and lag for a single MPC, indexed by $$k$$.
Each individual correlator output in a rake receiver is called a "finger". In direct sequence spread spectrum (DSSS), which is the spread spectrum method that 3G CDMA cellular systems use, the transmitted data stream is modulated by a "chip sequence" whose symbol rate is $$N$$ times higher than the symbol rate of the underlying data stream. This has the effect of "spreading" the signal energy over a bandwidth that is $$N$$ times larger than the underlying data stream. Each finger of the rake receiver then correlates against the chip sequence to "despread" the signal for a single delay value $$\tau_k$$. Because the chip sequence has $$N$$ times better time resolution than the data sequence, it can be used to measure $$\alpha_k$$ and $$\phi_k$$ for MPCs whose delays $$\tau_k$$ are fractions of a symbol period. In other words, each finger of the rake "resolves" a single MPC. Known pilot sequences are transmitted at regular intervals (i.e. different values of $$t$$), allowing for estimation of $$h(t, \tau)$$.