# 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.

As far as references on LTV system theory, I was first introduced to it via the text Wireless Communications, by Andrea Goldsmith. Chapter 3 of that book gives a pretty comprehensive overview of the topic. If you are feeling more brave, you can pull Bello's original paper on the topic where most of the terminology for basic LTV system theory was laid out (which usage persists to this day). A more recent reference by Matz expands upon the non-WSSUS case.

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 $$h(t, \tau)$$.