# Why use convolution for pulse compression in radar instead of correlation?

I have not understood the way to process pulse compression in radar signal processing. I think it's correct to use correlation for pulse compression, but I saw many materials about pulse compression. Then, they use convolution.

These drawings are my thinking. When I use correlation in first drawing. The output signal has max power. Because Transmit Signal is similar to Receive Signal. But when I use convolution, receive signal reverses. So I think this output signal power is smaller than output signal's power of first picture.

Using Correlation:

Using Convolution:

• Correlation is just backwards convolution, isn't it? Mar 5, 2020 at 17:29
• yeah right, but I think they make other result. Mar 6, 2020 at 3:25
• researchgate.net/profile/Ravibabu_Mulaveesala3/publication/… Mar 6, 2020 at 6:05
• I saw this pic. but I didn't understand. because the reference signal is not similar to receive signal. because when to use convolution, the receive signal or reference signal have to reverse. Mar 6, 2020 at 6:23
• The result is the same. Correlation is just backwards convolution (or convolution is just backwards correlation). dsp.stackexchange.com/questions/27451/… Mar 6, 2020 at 10:50

Let's start by getting out of the way the basic expressions and ideas for convolution and correlation.

## Convolution

For an input signal $$\x(t)\$$ going through a system $$\h(t)\$$, the output $$\y(t)\$$ is given by

$$y(t) = x(t) * h(t) = \int_{-\infty}^{\infty}x(t - \tau)h(\tau)d\tau = \int_{-\infty}^{\infty}x(\tau)h(t-\tau)d\tau$$

The engineering convention is usually represented by the most-right hand side. Of course, they are equivalent because convolution is commutative. However, my opinion is that the left hand side allows for a more intuitive explanation when considering signals passing through a system:

$$y(t) = \int_{-\infty}^{\infty}x(t - \tau)h(\tau)d\tau$$

Using the linear-time-invariant (LTI) concept, this describes that for every time shift $$\\tau\$$, the shifted version of the input signal $$\x(t-\tau)\$$ is weighted by some value given by the impulse response $$\h(\tau)\$$, which we then accumulate via the integral. It's important to see that the output of the convolution operation is a function of $$\t\$$. The variable $$\\tau\$$ is just a dummy variable used to calculate the integral and has no real meaning.

## Cross-Correlation

When doing correlation, we want to answer the question "how alike are two signals, $$\x(t)\$$ and $$\h(t)\$$, if I shift one of them by some delay $$\\tau\$$ for all time delays of interest?". This gives us a function of $$\\tau\$$ given by

$$C(\tau) = \int_{-\infty}^{\infty}x(t)^*h(t + \tau)dt$$

See now how the variable of integration is $$\t\$$ whereas for convolution it was $$\\tau\$$. Here, the variable $$\t\$$ has no real meaning since we're only concerned about the cross-correlation function being a function of the time delay only, which is relative. Nevertheless, we see that the two expressions are extremely similar.

If we cross-correlate the same function, then the equation becomes

$$R(\tau) = \int_{-\infty}^{\infty}x(t)^*x(t + \tau)dt$$

This gives us the definition of the autocorrelation $$\R(\tau)\$$ of $$\x(t)\$$.

## Matched Filter Theory

Matched filter theory has the result that the optimal filter, let's call it $$\h(t)\$$, that achieves the maximum signal-to-noise ratio (SNR) for a signal $$\x(t)\$$ after some delay $$\t_0\$$ is given by

$$h(t) = x(-t + t_0)^*$$

We see that the matched filter is the time-reversed complex conjugate of the input signal shifted by some delay $$\t_0\$$. This matched filter achieves the maximum SNR at $$\t = t_0\$$. In radar applications we're looking for the time delay of the target, so of course we don't know a priori what the delay will be to define the matched filter. It's possible to have multiple matched filters tuned for different $$\t_0\$$, but this becomes increasingly impractical to implement in a radar system.

A practical choice would be to set $$\t_0 = 0\$$ so that the new matched filter has a maximum SNR at $$\t = 0\$$. This way we need to only define one matched filter. We pay the price with potential SNR loss for other values of $$\t\$$. The new matched filter is then

$$h(t) = x(-t)^*$$

If we use this new $$\h(t)\$$ in the definition of the convolution integral we get

$$y(t)= x(t) * h(t) = \int_{-\infty}^{\infty}x(\tau)x(t + \tau)^*d\tau$$

If you compare this with $$\R(\tau)\$$, they are equivalent with the difference being that the conjugates are on the opposite functions thus changing the direction of the phasor rotations, which is usually of little consequence.

You can now see that computationally the convolution and autocorrelation functions are the same. The difference is the choice for $$\h(t)\$$, which is now the time-reversed complex conjugate of the signal you wish to receive.

Thinking graphically, since the signal which in our case is really the system $$\h(t)\$$ is already time-reversed, performing convolution flips the signal to its original orientation and you actually are now doing correlation.

The frequency sweep pulse compression: Both generating the transmitted pulse and performing the compression in the receiver is possible and was also practical with analog circuits. Chirping radar was popular before real time digital signal processing with tens of MHz sample rate. The special filter needed was a dispersive delay line which caused steeply frequency dependent time delay.

As already commented a pulse compression correlator can be considered to be a linear digital filter and that can be calculated by applying convolution with the reversed transmitted pulse in the baseband. The result is a short spike if the pulse do not autocorrelate.

There's a trap:

Correlation calculations take into the account the possible DC component by subtracting it at first. In statistics correlation of the processes occurs as coincident polarities after the averages are subtracted. Often the integration result is also divided by both RMS voltages, but that's only scaling.

Convolution in the receiver with the reversed baseband transmitter pulse (=matched filtering) should also be performed after the DC is removed (=subtract the averages). Otherwise there's at least something matching all the time - the DC and that spoils the detection spike.

That was my attempt to be intuitive. Unfortunately the exact presentation of the sameness must be mathematical. Learn the formulas of matched filtering and cross correlation of signals.

• As I mentioned above, I don't understand this example. Could you describe about this example? Both Transmit signal and receive signal are {3,2,1} that is right triangle shape. In convolution, Transmit signal is {3,2,1} and Receive signal changes to {1,2,3} because of reverse. Then, The output result is {9,12,10,4,1}. In correlation, Transmit signal is {3,2,1} and Receive signal is not changed. So It is {3,2,1}. Then, The output result is {3,8,14,8,3}. Their peak value and position are difference I think. Mar 7, 2020 at 5:29
• @jameshyun The answer is augmented
– user136077
Mar 7, 2020 at 10:13