# 6.2 Approach for range

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## Chirp and chirp train design

First, we design a Linear Frequency Modulated (LFM) chirp for the desired time-bandwidth product amount (TW), oversampling amount p, and sampling frequency (fs) . The MATLAB program" dchirp2 "will build up a single pulse while" ctbuild4 "will design a burst waveform consisting of L chirps, repeated over a period M.

## Example of"Dchirp2"Matlab function call

[s,h,y,T,W,Ts] = dchirp2(TW,p,sampfreq) with outputs:
• s = single LFM chirp set to specified input paramters
• h = match filter impulse response
• y = (used for testing of original code) passing s through match filter
• T = time duration of LFM chirp (units: sec)
• W= swept bandwidth of LFM chirp (units: Hz)
• Ts= sampling period (1/fs)

## Example of"Ctbuild4"Matlab function call

[s,ssent,h,y,T,W,Ts] = ctbuild4(TW,p,sampfreq,M,L) with outputs:
• s = single chirp defined by TW,p
• ssent = noise-free burst waveform of L lfm chirps of the same TW, p
• h = match filter impulse response
• y = (used for testing of original code) passing s through match filter
• T = time duration of LFM chirp (units: sec)
• W= swept bandwidth of LFM chirp (units: Hz)
• Ts= sampling period (1/fs)

In order to simulate proper radar returns, we first simulate the noise of a channel. Thus, we add complex white Gaussian nosie, using the"crandn"command in MATLAB, to our complex train of chirps. We perform this opertion in our main MATLAB function called" burst4 ."Next, we then apply a time delay to our signal. The program accomplishes that by shifting a vector x of length N to the right by amount TD.
For the project we worked only with an assumed value of SNR = -10 dB (i.e. the std. deviation is the square root of 10 multiplied against the chirp's average amplitude). We did a visual comparison of these two values to make sure our value of n was correct. See MATLAB function" burst4 "for detailed commentary.
The TD value inputted by the user is in terms of how many elements the user wants to physically shift the transmitted wave form by and do not correspond to actual units of time.

## Defining max range of radar processing

The Maximum Range of our radar system is determiend by two things. First, how large the resting time is between consecutive falling and rising edges of chirps. Secondly, how long our signal is. The reason why these are important is that since this model for range processing is in discrete time, we can not simulate an infinite range for our targets. That would correspond to having to time delay our transmitted signal by a extremely large amount. If shifted by an amount greater than the signal's length, the simulated return would just be a flat line of value zero for the length of the signal. Basically, the signal can be time delayed only up until the first LFM chirp's falling edge. This location corresponds to the last value of the recieved signal to ensure getting a big enough spike in the match filter's output to use in calculating range. Thus, since we define the value for time period (Ts) of our chirp pulse train and the number of pulses, we can define the max range by finding the maximum amount we can shift our signal. Then by plugging in this max Td value into the discrete-time range equation below, the max range value is found.

## Equation for max time delay

$\mathrm{MaxTD}=ML-M-N$

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