A linear frequency modulated chirp signal for radar is defined by the equation (1)

s(t) = exp(j*W/T*t^2)

*t = time* on the range of [-T/2,T/2], *T* =time duration in seconds of LFM signal pulse, *W* = swept bandwitdth over the life of the pulse in Hz

The changing frequency of the chirp signal sweeps from (-1/2)W to +(1/2)W Hz. It is interesting to note that the phase of s(t) varies quadratically versus t while the frequency changes linearly versus time. The deriviative of phase determines the instantaneous frequency of the signal. The signal is complex valued in this case because it is the baseband form of the linear frequency modulation. See Figure(1.1 ) below for example.

Thus, it would seem like the frequency spectrum S(f) would have most of its energy in the range of | f | < (W/2). However, this is in fact only true if the frequency sweeps slowly enough or if T is large enough.

In Matlab however, the chirp signal has to be represented as a discrete time signal. A solution to this problem is to just oversample s(t) enough so that we can effectively simulate the continuous time version. Otherwise, if we just wanted to have a discrete time signal, the sampling frequency would be kept approximately equal to the swept bandwidth W. But, we will need to oversample by at least a factor of 5 at least in order to properly simulate the continous time signal.

Since the chirp signal is compex valued and will be processed through a complex valued matched filter, all plots must be made of either the real part of equations (1), (2), or of the magnitude of the Fourier transform below

The sampling frequency* fs = 1 / Ts*. The sampling rate can also be tied to W, the swept bandwidth of the chirp, because in many cases the chirp is more or less bandlimited to a frequency extent of W. Therefore, it is convenient to let *fs = p*W* where *p>(or equal to) 1 *represents the *oversampling factor*. Thus, the equation for our discretized chirp changes into equation (2).

s[n]= exp[j*pi*W/T *(n*Ts - T/2)^2]

with 0<(or equal to) n < T. It is interesting to note that it may not be possible to make the discrete-time chirp symmetric, depending on how the sampling times are defined.

The magnitude of the Fourier transform of a chirp can be approximated by a rectangle in frequency if the time-bandwidth product is large (i.e., TW > 50 ).

Time bandwidth product (*TW*) is just the product of the chirp pulse's length *T* (units generally in microseconds) and the chirp's swept bandwidth *W* (units generally in Mhz). Thus the product itself should have dimensionless units.

If we assume that s(t) is the chirp and S(f) is its Fourier transform, we can approximate |S(f)| with a rectangle that extends from f = -W/2 and f = +W/2. Figure below shows that the rectangle approximation is not too bad. See Figure (1.2)

The Fourier Transform of a continuous-time chirp. Notice that most of the energy is concentrated in the frequency region between -W/2 and +W/2 (i.e. -1 MHz and +1MHz). The dashed line is the magnitude of an approximate transform |S`(f)|, which is perfectly bandlimited.

Moreover, the DFT of a chirp sometimes has interesting properties when p = 1. Since there are N nonzero samples in a discrete time chirp signal, the N-point DFT can be taken. Plus, whenever N equals a multiple of 4 and p = fs/W = 1, the DFT S[k] is also an exact chirp.

The transmitted signals are designed differently for range and velocity processing. In the case of range processing, the output SNR and range resolution must be maximized. Thus, LFM chirp signals with large TW products are used in conjunction with a "pulse compression" *match filter*.

The actual transmitted waveform for range processing for our project will be a burst waveform of consecutive LFM chirp signals of the same TW, p paramters. The reason why is because of taking into account possible future work on this project. If the range and velocity analyzers were ever to be integrated, a burst waveform of LFM chirps of the same p, TW paramteters would have to be used as the transmitted signal for both. Thus, having the range analyzer work with such a signal already facilitates the possibility of integrating the two systems.

When used with each other, the SNR is maxized and the detectability of changes in our signal is enhanced. The resulting output of our match filter, LFM chirp combination is a very narrow "compressed" peak that has a large amplitude relative to the rest of output. This makes the detection of echoes easier, and more on how exactly is discussed in the "Approach for Range" module. In the case of velocity processing, a radar relies on Doppler frequency shift caused by a moving target. The processor must perform a spectrum analysis, and the magnitude of the Doppler shift is so small that a different type of transmited waveform is needed - a single pulse would not work. A

*burst waveform *consisting of a coherent group of short pulses is sufficient enough to be used. See the "Approach for Velocity" module for details.