# How to get the channel impulse response by using PRBS signal with BPSK modulation?

I want to get the channel impulse response by sending PRBS. Following is some channel measurement flowchart that I found

a(t) is the PRBS，sending it to transmitter to upconversion to BPSK modulated signal s(t)，r(t) represnet the receiving signal that pass throung real wireless channel. The receiver would downconversion the receving signal r(t) into I(t) and Q(t) signal. After that, taking a(t) and I(t)+jQ(t) to do cross correlation to get the channel impulse response.

Yet, there is still some point that I confused about：

1. As far as I know, BPSK modulate and demodulate only need one I(t) signal，why the receiver output I(t) and Q(t) two signal?

2. After downconversion r(t), we get I(t)、Q(t) baseband signal. I am wondering aoubt the form of I(t)、Q(t) signal if I represnet a(t)、s(t)、r(t) as follow.

the point that I got stuck in is that is I(t) Q(t) signal BPSK demodulated signal or not? Moreover, no matter I(t) Q(t) is demoudulated signal or not, I both have problem. I describe it as below

• Case 1： if I(t) Q(t) is BPSK demoudulated signal

The channel would definitely caused delay which would make the signal with phase shift that system unknown, so how can receiver demodulate the signal accurately?

• Case 2：no need to do demodultion at Rx

To the best of my knowledge, if I have PRBS a(t) and the receving PRBS a(t-delay),then, after cross correlation of two signal I can get the delay time. Howerver, what if the receiver didn't demodulate the signal, how can I get the delay time and channel impulse response? Say the problem in another way is that, why calculate corss correlation of a PRBS signal and a sinusoid(I+jQ) signal can get the channel impulse response?

Do not think of your transmitter as BPSK since you are not intending to do data transmission (as you will not be doing data demodulation in the receiver) but simply an upconverter of a bipolar waveform (yes as far as the trasmitter operation goes, this is indeed identical to BPSK but I think that is giving you much confusion). So the transmitter is frequency shifting the baseband signal to an arbitrary carrier frequency.

The receiver simply down-converts the signal back to baseband.

Because of the phase and frequency offsets (in addition to the multipath distortion) the signal as received will no longer exist at baseband as a real signal but will have real and imaginary components. We need to maintain all of these components in order to solve for the channel. Therefore you must do a quadrature down-conversion that would maintain the real and imaginary components (I and Q) of the complex waveform. A very simple example is a channel with an impulse response represented by an impulse at t = 1 sec with a phase of 45°.

$$c(t) = \delta(t-e^{j\pi/4})$$

The received signal is the convolution of our transmitted signal with the channel, so in this case the BPSK transmitted signal would be received as a complex having a 45° phase rotation:

This is a very simple channel; in practice you will have multiple delayed copies at different delays and amplitudes combining in a possibly unrecognizable pattern in the receiver. If the transmitted waveform and the resulting received waveform is known, then you can use inverse convolution techniques such as the Wiener-Hopf equations to solve for the unknown channel, I describe this in detail in the following post:

Compensating Loudspeaker frequency response in an audio signal

(For the case of solving for the channel instead of the channel compensator, swap Tx and Rx as described in that post)

A PBRS (Pseudo-Random Binary Sequence) is an excellent choice for a training sequence as it has spectral content across the entire band. The equalizer can only determine its compensation at frequency locations where energy is present.

• Thank you for your detailed elaboration. I really appreciate it! – Y.Y.Lin Nov 13 '19 at 5:11