The spectrum of a chirp pulse describes its characteristics in terms of its frequency components. This frequency-domain representation is an alternative to the more familiar time-domain waveform, and the two versions are mathematically related by the Fourier transform. The spectrum is of particular interest when pulses are subject to signal processing. For example, when a chirp pulse is compressed by its matched filter, the resulting waveform contains not only a main narrow pulse but, also, a variety of unwanted artifacts many of which are directly attributable to features in the chirp's spectral characteristics.
A simple way to derive the spectrum of a chirp using a computer is to sample the time-domain waveform at a frequency well above the Nyquist limit and use an FFT algorithm to obtain the desired result. As this approach was not an option for the early designers, they resorted to analytic analysis, or and to graphical or approximation methods. These early methods still remain helpful, however, as they give additional insight into the behavior and properties of chirps.
Chirp pulse A general expression for an oscillatory waveform, centered on frequency is
where and (t) give the amplitude and phase variations of the waveform , with time. The frequency spectrum of this waveform is obtained by calculating the Fourier Transform of , i.e.
so
In a few special cases, the integral can be solved to give an analytical expression, but often the characteristics of and (t) are such that the integral can only be evaluated by an approximation algorithm or by numerical integration.
Linear chirp In the special case where s(t) is constrained to be an up-chirp, flat topped pulse with its instantaneous frequency varying as a linear function of time, then an analytical solution is possible.
For convenience, the pulse is considered to have unit amplitude and be of duration T, with the amplitude and phase defined over the time interval -T/2 to +T/2. The total frequency sweep is F, varying in a linear manner from -F/2 to +F/2 in the defined time interval.
When the frequency is a linear function of time, the phase is a quadratic function, and s(t) can be written
The spectrum of this linear FM signal is
By completing the square and using the Fresnel integrals C(X) and S(X),Jahnke E. and Emde F., "Tables of functions", Dover Publications N.Y. 1945Abramowitz M. and Stegun I.A.,"Handbook of Mathematical Functions", Nat. Bur. Standards 1964, reprinted by Dover Publications N.Y. 1965 (9th ed.1972) defined by
the expression can be evaluatedKlauder J.R., Price A.C., Darlington S. and Albersheim W.J., The Theory and Design of Chirp Radars", The Bell system Technical Journal, Vol.39, July 1960 (pp.745-809)Chin J.E. and Cook C.E., The Mathematics of Pulse compression", Sperry Eng. Review, Vol.12, Oct 1959. (pp.11-16)Cook C.E., Pulse Compression - Key to More Efficient Radar Transmission", Proc.IRE, March 1960 (p.312)Cook C.E. and Bernfeld M., "Radar Signals - An Introduction to Theory and Application", Academic Press 1967,1987, reprinted by Artech House 1993.Varhney L.R. and Thomas D.,"Sidelobe Reduction for Matched Filter range Processing", IEEE Radar Conference 2003 to give:
where and are given by
The linear FM spectrum can be considered to have three major components, namely an Amplitude Term, a Square Law Phase term, and a Residual Phase Term
The ratio is approximately unity over a large part of frequency range of interest so approximates to a constant phase angle /4 there. If a frequency scaling term n is introduced, where , then the expressions for the Fresnel arguments become and
The spectra are now functions of the product T.F, independent of any particular values of center frequency and bandwidth. This product, T.F, is often referred to as the time-bandwidth product of the chirp.
Tables of the Fresnel integrals have been published, together with mathematical routines with which to compute the integrals manually or by means of a computer program. In addition, a number of mathematical software programs, such as Mathcad, MATLAB and Mathematica have built-in routines to evaluate the integrals, either as standard functions or in extension packages.
Some plots of the power spectrum |S()|2 as a function of frequency are shown, for time-bandwidth products of 25, 100, 250 and 1000. When the product is small, the Fresnel ripples are very much in evidence, but the spectrum does tend to a more rectangular profile for larger values.
In the case of the plots of residual phase, 2(), the profiles tend to be very similar over a wide range of time-bandwidth products. Two examples, for TxB = 100 and 250 are shown below. They have a phase angle close to a value of /4 within the chirp range and they only start to change significantly for frequencies beyond this range.
Consequently, for frequencies within the sweep range of the chirp, it is the square-law phase term 1() and its group delay function ( = -d1/d() ) that are of most interest. There is a plot of the group delay shown below. Both this function and the phase 1() are independent of the value of the time-bandwidth product. As expected, the group delay is a linear function with a duration T secs, over a frequency sweep of rads.
The residual phase term adds only minor perturbations to this characteristic within the frequency range . At frequencies outside this range, 2() deviates rapidly from /4, and so the total phase will deviate seriously from a square law there. Fortunately, the energy content of the chirp spectrum is very small at these frequencies (as is demonstrated in a later section).
Non-linear chirps When the Frequency-Time characteristic is non-linear, the Fourier integral is difficult to evaluate. In such cases, it is possible to resort to an approximation method such as the stationary phase approximation, or to use numerical methods.
Via stationary phase method Often (as in radar applications) a(t) is a slowly varying function of time and the phase (t) is oscillatory and varies rapidly, over the range of integration. With such waveforms, the stationary phase approximation can be used to investigate the spectrum.Fowle E.N., "The design of FM pulse compression signals", IEEE Trans. IT-10, 1964, (pp.61-67)Key E.L., Fowle E.N., Haggarty R.D.., "A method of pulse compression employing nonlinear frequency modulation", M.I.T. Lincoln Lab., Lexington, Mass., Tech. Rep. 207, 1959.Key E.L., Fowle E.N., Haggarty R.D.., "A Method of Designing Signals of Large Time-Bandwidth Product", Proc. IRE Int. Con. Rec. Pt.4, March 1961 (pp.146-154) The method relies on the fact that the major contributions to the Fourier integral come from the region where the rate of change of phase is minimal, i.e. when
Unless (t) is a constant, the point in time ts at which the phase is stationary will vary according to the instantaneous frequency . Expressing the difference between (-).t and (t) as a Taylor series about the time ts, but discarding all but the first three terms (of which the second term is zero, here), the Fourier integral can be written, approximately, as
In this equation ts represents a constant time point, so terms depending on ts alone may be taken outside the integral. The expression simplifies to
so
where is used to indicate the dependence of the frequency variable on t. This is a very useful expression linking, as it does, the spectrum profile to the amplitude and phase characteristics of the chirp.
To carry out the inverse process, i.e. to find the time domain function s(t) given frequency domain data, the inverse Fourier transform is derived.
where (x) is the phase function of the spectrum. The stationary phase points for this integrand are located at
and the corollary relationship, equivalent to that derived for the spectrum, can be obtained by the stationary phase method, and is
In effect, stationary phase analysis gives the following (approximate) Fourier pair relationships:
and
Consequently, approximate expressions for a(t) and (t) can be obtained when the spectrum, including its phase function () is given and, similarly, approximate expressions for |S(| and () can be obtained when the signal characteristics are given. Several examples of the procedure are given in the literature
Although the relationships are only approximate, their accuracy improves as the time-bandwidth product increases. In cases where the signal envelope and spectrum modulus are defined by smoothly varying Gaussian function then a T.F product as low as 15 will give acceptable results, but if both a(t) and |S()| are defined by rectangular functions, then the product T.F needs to be much greater, typically over 100.
Examples Typically, in the radar case, a(t) is a constant over the duration of the signal and, for convenience, is assumed here to be unity. So the phase and amplitude characteristics, in the frequency domain, are related by
There are two solutions for (), which are complex conjugates of each other. The two filters with these characteristics can be used as the transmitter and receiver filters of a radar system and are interchangeable. The group delay characteristic D(), (where D()=-d/d), is
so
So in the case of a rectangular time envelope, the dispersive delay characteristic is given by the integral of the square of the envelope. If the positive sign is taken, then the group delay increases with increasing frequency and vice versa. The result is only approximate, but is more accurate for large values of the time bandwidth product. Consider, as an example, the case of a spectrum that is uniform over the range -/2 to /2, then
so
Put D(-/2) = 0 and D(/2) = T, where T is the pulse duration, then K = T/2 and A = (2πT)/ωmax so, finally
As expected, a flat topped frequency spectrum corresponds to a linear frequency sweep.
The linear chirp is just one special case which, in any case, can be calculated more precisely by the methods of the earlier section. The particular usefulness of the stationary phase method lies in its ability to provide results when the frequency sweep is non linear. In such cases the spectral response can be shaped to meet some desired design criteria, for example, low side-lobes when a chirp is compressed. One such family of spectral functions that has been studied is given by
It is possible to find the group delay characteristics of these functions in a similar manner to that carried out above and the results for n = 1 to 4 have been calculated. Although these cosine functions are amenable to mathematical manipulation, they are rarely chosen to define the spectral characteristics of a chirp, in practice, because when compressed they give broad main pulses with high side-lobe levels. A better characteristic (among many)Harris F.J., "On the Use of Windows for Harmonic Analysis with the Discrete Fourier Transform", Proc. IEEE Vol.66, Jan 1978 is the Hamming function, given by
A plot of this characteristic is shown, plotted over the range -/2 to /2.
Applying the equations given above, the group delay characteristic which achieves this spectral shape can be obtained. It is
Now because the principle of stationary phase shows that there is a direct relationship between elapsed time and the instantaneous signal delay then, for the Hamming window, t/T can be related to / by
This characteristic which is time as a function of frequency is shown here. Inverting the plot gives the more usual (and more useful) plot of frequency as a function of time, which is also shown.
Other spectral shapes can be investigated in the same way and the results, although approximate, are surprisingly accurate, especially when the time bandwidth product of the pulse is high.