WO1995032437A1

WO1995032437A1 - Signal processing apparatus and method

Info

Publication number: WO1995032437A1
Application number: PCT/GB1995/001163
Authority: WO
Inventors: Brent Summers
Original assignee: Commonwealth Of Australia
Priority date: 1994-05-20
Filing date: 1995-05-22
Publication date: 1995-11-30
Also published as: AU2532195A

Abstract

By employing waveforms having a waveform repetition interval and bandwidth related such that the product of the waveform repetition interval and the bandwidth is an integer, the performance and efficiency of both signal processing apparatus and methods can be improved, particularly where the waveform comprises a plurality of identical sweeps and each sweep has a waveform repetition interval and bandwidth related such that the product of the waveform repetition interval and the bandwidth is an integer.

Description

SIGNAL PROCESSING APPARATUS AND METHOD

This invention relates to a signal processing apparatus and method and in particular to

apparatus and methods for use in rf transmission systems operating with Frequency modulated

continuous wave (FMCW) type waveforms.

The invention enables a waveform, for use in a radar system for example, to be generated

which is optimized for peak power limited transmission systems by minimising the peak to mean

ratio whilst simultaneously providing exceptionally good out-of-band emission characteristics and conventional radar waveform properties. Consequently, the method is very suitable for use with non-linear transmission systems, for example those employing a high power amplifier in compression which are required to comply with tight out-of-band spectral emission specifications.

In a conventional radar transmit sub-system FMCW waveform generation can be accomplished by either direct rf synthesis or baseband synthesis followed by up conversion. Either technique can be implemented in analogue or digital technology. The disadvantages of rf synthesis are two fold. Firstly, the waveform flexibility is limited by either the precision in the analogue components, for analogue systems, or the computational rate for digital systems.

Secondly, the dynamic range and/or linearity of such implementations are frequently technology

limited.

The disadvantage of baseband synthesis followed by up-conversion is that the precision ~>

generated in a baseband synthesized waveform is in practice difficult to preserve in the up-

convertion and amplification stages because analogue mixers are by definition non-linear

devices and for reasons of efficiency high power amplifiers often operate with some

compression.

The problem therefore is how to preserve the fidelity of a precision generated waveform through a system with a non-linear transfer function, when, as is well known, passage of a bandlimited signal through a non-linearity results in out-of-band emission and in-band distortion. Out-of-band emission wastes power and causes interference with other users of the band and often international regulations governing the extent of this exist while in-band distortion results in a degradation or loss of the waveforms properties.

Fundamentally, an ideal transmitted waveform must be simultaneously power efficient, bandwidth efficient and preserve the integrity of the general waveforms natural properties. To explain these subtle ideals a little further:

A maximally 'power efficient' waveform is one in which the ratio of the peak to mean temporal magnitude envelope is unity.

A 'bandwidth efficient' waveform is one in which all of the power transmitted is uniformly 'contained' within the sweep/receiver bandwidth. Power transmitted out-of-band may not only

interfere with other users but, from the radar point of view, is a system loss - since it serves no useful purpose whatsoever in the process of target detection. Preservation of a waveforms natural properties is taken to mean minimisation of in-band distortion for both amplitude and phase. In this way, a waveform which has very good

correlation properties is not compromised by the transmission path. It is often the case that a

waveform with, say, very low time-sidelobes in correlation is very sensitive to phase distortions

or bandwidth mismatch.

This invention provides a signal processing method and apparatus for producing and

employing waveforms coming closer to this ideal than has previously been possible.

In a first embodiment this invention provides signal processing apparatus employing a waveform having a waveform repetition interval and bandwidth related such that the product of the waveform repetition interval and the bandwidth is an integer.

Preferably the waveform comprises a plurality of identical sweeps, each sweep having

a waveform repetition interval and bandwidth such that the product of the waveform repetition interval and the bandwidth is an integer and may further comprise a series of dwells each comprising a plurality of identical sweeps.

Advantageously a window function can be applied at the start and finish of each dwell

in order to remove the boundary discontinuity at the start and finish of the dwell. A suitable

window function is a Tukey window function.

Advantageously the frequency spectrum of the waveform can be weighted, preferably by applying a window function such a Tukey window function to it. In a second embodiment this invention provides a window function comprising a cos"x

75 function in which n is non zero and positive convolved with an extending function having a

constant non-zero value across a single continuous range and a value of zero elsewhere.

In this case a window function Wz can be used defined by the equation;

80 where

W. = ∑ G E_z___χ : z = (x+y),...,(x+y)

Cg

G = cos x = -X,..._X n≥O

1 +Jfc

Ey = 1 :y = -Y,...,7

J z=-(x+r)

Preferably the window function is the square root of the result of the convolution. In this case a window function Wz can be used defined by the equation; 5

where W = ∑ G E___χ :z= -(x₊y),...,(x₊y) θr

" Y-τ

Q = cos" — : x = -X,...J( n≥O

_Ey = 1 y = -Y,...,Y

g Σ ^wz

A particularly advantageous arrangement of signal processing apparatus provided by the first embodiment of the invention is to weight the waveform frequency spectrum in such signal

90 processing apparatus with a window function provided by the second embodiment of the invention. When this is done the value of the index n in the window function can be selected to minimise the peak to mean ratio of the waveform, the preferred way of doing this being to select the value of the index n such that the first and second Fresnel peaks in the waveform envelope magnitude function have equal magnitude.

95

In a third embodiment this invention provides a method of generating a waveform by

defining the waveform repetition interval (T) and bandwidth (B) such that T.B. = an integer.

Preferably the waveform is generated by then evaluating the equation

g(t) = G_n.e *^■«* * bn -i* c)

100

where M = (TB -l)/2

Advantageously the variables a, b and c can be defined as;

a = WTB

b = IWtlT)

105 and

— [-\ ^(ra+1) + 2TB + 1] for phase continuity (TB odd case only) 4

In a fourth embodiment this invention provides a signal processing method employing a waveform having a waveform repetition interval and bandwidth related such that the product of the waveform repetition interval and the bandwidth is an integer.

110 Preferably the waveform comprises a plurality of identical sweeps, each sweep having a waveform repetition interval and bandwidth such that the product of the waveform repetition

interval and the bandwidth is an integer and may further comprise a series of dwells each

comprising a plurality of identical sweeps. 115 Advantageously a window function can be applied at the start and finish of each dwell

to remove the boundary discontinuity at the start and finish of the dwell. A suitable window function is a Tukey window function.

Advantageously the frequency spectrum of the waveform can be weighted, preferably

120 by applying a window function such a Tukey window function to it.

It is particularly advantageous for a signal processing method according to the fourth

embodiment of the invention to weight the waveform frequency spectrum with a window

function provided by the second embodiment of the invention.

125

Apparatus and methods employing the invention will now be described by way of example only with reference to the accompanying diagrammatic figures in which;

Figure 1 shows the frequency spectrum of a typical conventionally generated waveform;

130

Figure 2 shows the frequency spectrum of a waveform produced employing the

invention;

Figures 3 A to 3D show the frequency spectra of waveforms produced conventionally and 135 employing Tukey windows;

Figures 3E to 3H respectively show the frequency spectra of waveforms corresponding

to those in Figures 3A to 3D respectively and produced employing the invention and Tukey windows;

140

Figures 4A to 4D are explanatory diagrams showing Tukey dwell weighting functions;

and

Figures 4E to 4H respectively show the waveform spectral occupancies produced by the

145 Tukey dwell weighting functions of Figures 4A to 4D respectively.

Figures 5A to 5D are explanatory diagrams showing Tukey spectral weighting functions having different values of taper;

150 Figures 5E to 5H respectively show the waveform envelopes corresponding to the Tukey functions of Figures 5A to 5D respectively;

Figure 6 shows a full definition of the new window function according to the invention;

155 Figures 7 A to 7D show examples of the new window function for different values of n and having 100% taper;

Figures 7E to 7H respectively show the spectral magnitudes corresponding to the

windows shown in Figures 7A to 7D respectively; 160

Figures 8A to 8D show the new window function for different values of n with 50% taper; Figures 8E to 8H respectively show the spectral magnitudes corresponding to the

window functions shown in Figures 8A to 8D respectively; 165

Figures 9A to 9D show the cos^πTukey spectral weighting function with 20% taper for

different values of n;

Figures 9E to 9H respectively show the waveform envelope magnitude functions

170 corresponding to the spectral weighting functions of Figures 9A to 9D respectively;

Figures 10A to 10D show the square root cosTukey spectral weighting function for 20% taper different values of n;

175 Figures 10E to 10H respectively show the waveform envelope magnitude functions corresponding to the spectral weighting functions of Figures 10A to 10D respectively;

Figures 11 A to 1 IH show different waveform characteristics for an optimised square root cosⁿTukey spectral related waveform; 180

Figure 12 is a table of some optimum values of index n for various waveform parameters.

Figure 13 shows the spectral occupancy of an optimised waveform; and

185 Figure 14 shows signal processing apparatus for use in a radar system employing the

invention. In conventional waveform generation over a time interval (T) a bandwidth (B) is swept

in a linear manner, i.e. f = f₀ + Bt/T.

190 Since frequency can be defined as the rate of change of phase with respect to time, i.e. f = dφ/dt the phase φ of a time-domain signal g'(t) is obtained by integration.

Hence t g t) = A(t).e _o >

g'(t) = A(t).e J -- ^* <^{ra 2}> ^{• rø2} for t < 772

195 where, A (t) is some amplitude scaling factor (usually 1).

It is a popular misconception that the spectral occupancy of a linear frequency modulated waveform (LFM), commonly known as a chirp, contains just those frequencies of the swept bandwidth. It does not. Such a signal is a time-limited signal and it is a physical fact that 'if a 200 signal is time-limited it cannot be simultaneously bandlimited and vice versa'.

Consequently, the digital representation of a chirp suffers from aliasing unless

oversampled by a considerable amount.

205 A radar or sonar often uses a waveform termed a "dwell" and comprising a number of sweeps. In such a case the dwell waveform g"(t) (also time limited) is obtained by replicating the time-limited signal g'(t) at intervals of T

where * represents convolution

Ns = Number of sweeps in a dwell

T = waveform repetition interval (WRI)

Such a signal (depending on the values of T and B) may or may not be continuous, i.e. it may not or may posses discontinuities in either amplitude or phase or their derivatives at the

WRI boundaries.

In the optimum case, the waveform is continuous and as a result the frequency spectrum

or spectral magnitude has a line structure with a line spacing dependent on the dwell time rather than the WRI. However in the strict sense this is still not bandlimited as there are an infinite number of these lines. Hence the problem in digital representation. An example of such a frequency spectrum is shown in Figure 1.

In Figure 1 it can be seen that the spectral occupancy of the waveform extends far

beyond that of the design bandwidth B . It can be further seen that the spectral line peaks in-band are of different magnitude due to the Fresnel ripples. This also is undesirable because it increases the peak to mean ratio of the waveform, making it less power efficient. In the present invention the method of waveform definition and generation is completely

230 different from that of the conventional approach hereinbefore described, and is described below

as a series of steps. Although the greatest benefits can be obtained by use of all the steps it must

be emphasised that some benefit can be obtained by using only some or even only one of them

in isolation from the others.

235 Step 1

Define a relationship between the waveform parameters T and B, i.e.

T x B = integer In this way, the waveform is defined as true 'periodic bandlimited waveform' containing no discontinuities at the WRI boundaries. Such signals are entirely defined by a line spectrum

240 comprising a finite set of lines rather than an infinite set. Consequently, the waveform can be defined not only in the instantaneous frequency-domain (as is the case of the conventional chirp) but also in the normal frequency domain as comprising a finite set of unit spectral lines of 1/WRI spacing, that is spaced at the waveform repetition frequency, quadratically phased. As a result the waveform can be exactly represented by digital samples and the waveform can be precisely

245 defined in the frequency domain even though it is time limited. In order to exactly represent a conventional (non-bandlimited) chirp waveform digitally it would theoretically be necessary to

have an infinite sampling rate. In practice a sampling rate high enough to give acceptable results is used but a trade off between accuracy of representation and sampling rate must always be made.

250

The number of spectral lines in the design bandwidth is T.B. This is the only way of defining a waveform that is perfectly bandlimited and this provides great advantages in baseband synthesis because it allows the synthesised waveform to

255 be an exact representation of the desired waveform.

Step 2

The basic waveform is obtained in the time-domain by means of a inverse finite Fourier

series (or polynomial). That is to say:

260

M g(t) = (\-jTB). ∑ G_π.e ⁱ-^{an2 +} *" ^{+ c}>

»— M

where M = (TB-\)I2

a ~^~ WTB

b = 2U(t/T)

π TB*< c = — [-1 + 2TB + 1] for phase continuity (TB odd case only)

4 2

265

Since the above equation is expressed in terms of the continuous variable 't' it is valid for

all values of time.

Therefore a dwell waveform g" (t) can be evaluated directly without the need to replicate 270 a WRI waveform g'(t) as in the conventional case. Although this could be done if there was some practical advantage in doing so.

Furthermore, since the function is exactly bandlimited the approach lends itself to digital synthesis. 275

Other values for the variables a, b and c can be selected to allow other phase-coded waveforms to be generated by this method if desired.

Waveforms generated in this way can be 100% power efficient when they are not

280 amplitude modulated since they can have a peak to mean ratio of 1 and all of the transmitted

power can be contained within a set design bandwidth. It is possible to trade off bandwidth against power efficiency because as the bandwidth is reduced the power efficiency will drop and vice-versa since power transmitted outside the designed bandwidth is wasted in a radar or sonar

system. 285 Step 3

Digital baseband samples of the waveform are simply obtained by evaluating the above

equation at discrete values in time i.e. replace the continuous variable 't' with k .

290 So long as the sampling frequency V- is greater than the design bandwidth B, Nyquist's

rule is satisfied and the samples will be an exact representation, in the sampling theorem sense,

of the waveform. In contrast samples taken at the same rate of a conventional time-domain

generated chirp can never be an exact representation due to the infinite member of lines in the

frequency spectrum.

295

In radar and sonar systems the received and transmitted signals are correlated in order to identify where the received signal has been retumed from, the fact that the digital samples of the waveform produced according to the invention are an exact representation of the waveform improves the temporal robustness of the waveforms when correlated.

300

Figure 2 illustrates the basic spectral occupancy characteristics of the polynomal waveform evaluated from the above equation for TB = 5 and 1"^ = 8.

When compared with the conventional case of Figure 1 it can be seen that the number 305 of lines is finite as expected and the out-of-band emission is entirely determined by the sidelobes

of the sine due to the finite dwell. Furthermore the in-band lines are of uniform magnitude.

Unlike the conventional waveform the number of sweeps in a dwell directly affects the out-of-band emission levels since the width of the sine is inversely proportional to the dwell- 310 time.

Consequently, for a given sidelobe roll-off rate, a doubling in Ns will double the roll-off

rate. This is a very important feature since most practical radar or sonar waveforms comprise

a relatively large number of sweeps per dwell (typically 256) and as a result will automatically 315 have a very high sidelobe roll of rate, reducing out of band emissions.

Both Figure 1 and 2 illustrate a typical spectral template for out-of-band emission levels to allow simple comparison between the conventional and inventive waveforms.

320 Step 4

Since the out-of-band emission level is entirely determined by the finite dwell-time, further improvements in out-of-band suppression can be achieved by application of a window function across the dwell.

325 However, in order to minimise the weighting loss the essential requirement is to minimise the discontinuity at the start and end of the dwell. This < ai\ be accomplished by means of a Tukey window function, that is a window function having a raised cosine start and end taper of small percentage taper. Weighting loss is defined as;

330 Weighting loss (db) =

. _V-1

10 log-i ∑ W_k

N *₌₀ Figures 3A to 3D illustrate the effect of application of a 3,125% taper Tukey window on a conventional waveform for N_s = 2.4, 8 and 16 respectively while figures 3E to 3H respectively show the corresponding polynomial waveforms according to the invention. In the case of the

335 conventional waveforms it will be noted that the peaks of the out of band lines do not decay with

increasing Ns. Whereas in the case of the new waveforms the increased decay rate is dramatic. Thus the invention gives a very considerable improvement in out-of-band suppression for

minimal weighting loss, (.14db).

340 Since the raised cosine end tapers of the Tukey window place the discontinuity into the

second derivative the decay rate is .18dB/octave increasing by .6db/octave for each doubling of

Ns.

Whilst increasing the percentage taper also improves the situation this is considered

345 undesirable since the weighting loss will become more significant. The effects of increasing percentage taper are shown in Figures 4. Figures 4A to 4D show the dwell weighting functions for Tukey windows with 0%, 10%, 20% and 40% taper respectively while Figures 4E to 4H respectively show the corresponding waveform spectral occupancies for the inventive waveform with N_s = 4. 350

Step 5

Having improved the out-of-band emission levels by dwell weighting a polynomal waveform consideration is now given to simultaneously improving the peak to mean ratio.

355 In the time-domain the magnitude of g(t) contains the Fresnel ripples - as illustrated by Figures 5E to 5H.

These ripples (in excess of 3db relative to the mean) for a peak power limited system are

a problem since to avoid saturation (or clipping) it is necessary to reduce the effective

360 transmitted power.

Consequently, it is very desirable to minimise this ratio by some means. It has been found that the application of spectral tapering has this effect.

365 Figures 5 illustrate the effect of application of a Tukey windows of 0, 10, 20 and 30% taper on the design spectral magnitude {Gn} on the time-domain amplitude envelope. Figures 5A to 5D show the Tukey spectral weighing functions and Figures 5E to 5H the corresponding time domain waveform envelope magnitude functions for Tukey windows of 0, 10, 20 and 30% taper respectively. It can be seen that as the % taper is increased there is a corresponding

370 decrease in the peak-to-mean ratio.

Step 6

A further extension of this idea is to change the raised cosine shape of the end taper so as to minimise the effect of the taper on the in-band signal.

375

In order to do this a new window function has been invented dubbed co^ - Tukey.

This window function is generated by convolving a cos"x function with what is termed an 'extending function' and a full definition of the new window function is given in Figure 6. 380 The ratio of the number of elements in the generating window to extending function determines the % taper whereas the index 'n' determines the shape of the taper.

It will be realised that the cos" -Tukey window function can be used generally in any 385 application where window functions are employed, but as will be explained it is particularly

advantageous in conjunction with waveforms produced by the above method.

Figures 7 A to 7D illustrate the cos" -Tukey window with 100% taper for n = 0, 1, 2 and

3 respectively while Figures 7E to 7H respectively show the corresponding spectral magnitudes.

390

It will be noted that when n = 0 a triangular window results.

When n = 1 a raised consine window results. And for every integer increase in 'n' the sidelobe roll-off rate increases by .6dB/octave.

395

Figures 8A to 8H illustrate the same scenario as in Figures 7 for the cosⁿ-Tukey window with 50% taper.

Figures 9A to 9D show the cos"Tukey spectral weighting function with 20% taper for

00 n=0, 1, 2 and 3 respectively while Figures 9E to 9H respectively show the corresponding

waveform envelope magnitude functions.

Application of cos"Tukey window to the design spectral magnitude for a fixed 20% taper and n = 0, 1, 2 and 3 is shown in Figures 9. It can be seen that the shape of the window affects

405 the peak-to-mean ratio as well as the sidelobe roll of rate.

A further step in this process is to take the square root of the window to produce a window function dubbed the square root cos" Tukey window function. The effect of this is

shown in Figures 10. Figures 10A to 10D show the square root cos" Tukey spectral weighting

410 function with 20% taper for n = 0, 1, 2 and 3 respectively while figures 10E to 10H respectively show the corresponding waveform envelope magnitude functions.

This has the effect of leaving more energy in-band but it also reveals a feature which is further exploited. If Figure 10E is examined it can be seen that a second peak emerges which 415 in this case is larger than the first. In Figure 10F when n=l this second peak is still visible but is now smaller than the first.

From this observation is concluded a very important point:

420 Minimum peak-to-mean ratio occurs when the size of the first and second amplitude envelope peak are equal. In other words the energy in the first peak which dominates the peak- to-mean ratio is equal split between two peaks. For the case illustrated in Figures 10 this implies that there is an optimum value of 'n' which exists somewhere between 0 and 1.

425 It turns out that it exists at n = 0.4601. Figures 11 illustrate the complete waveform

characteristics for this case. With reference to Figures 11 ; 11 A Design spectral magnitude with 20% square root cos " ⁼ ° ⁴⁶⁰¹ - Tukey Window.

1 IB Design quadratic phase.

11C Amplitude envlope of waveform in the time-domain illustrating equal peaks.

1 ID Corresponding time-domain phase of waveform - notice also quadratic.

1 IE Real and Imaginary components of waveform illustrating generated chirp.

1 IF Corresponding instantaneous frequency response.

11G Circular correlation function of waveform

- Note that since it was a window applied to the spectral magnitude in waveform design the circular correlation function is the transform of window. Hence .18dB/oct are time sidelobe roll off rate. 1 IH Dwell weighting function 3.125% Tukey.

The optimum value of the index 'n' to minimise peak to mean ratio is a function of TB and the percentage spectral taper and can be computed beforehand so that it can be used as a look-up table or could be calculated in real time for each desired waveform. Figure 12 is a table of some o timal values of 'n' found by computer optimisation for TB's in the range 20 to 100 for the case of 30% spectral taper.

Figure 13 illustrates the spectral occupany of an optimized waveform for TB=21, Ns =

64. It can be seen that in comparison to the conventional chirp illustrated in Figure 1 there is

a dramatic improvement in the out-of-band emission levels. Minimum peak-to-mean (i.e. <0.9dB) provides an optimal compromise between bandwidth efficiency and power efficiency and minimises the effect of non-linearities. n~>

Figure 14 shows signal processing apparatus for use in a radar system employing

waveforms produced using the techniques described above. The waveforms are produced by

a digital waveform generator 1 and supplied as a series of baseband samples spaced to form an

455 exact replica of the waveform to an upconverter and drive 2. The r.f. signal from the upconverter 2 is supplied to a high power amplifier 3 which is connected to an antenna 4 for

transmission.

In practical apparatus of this type the upconverter 2 and amplifier 3 will be non-linear 460 and the use of the inventive waveform definition techniques described above allow waveforms minimising the effects of these non-linearities to be produced.

Claims

1. Signal processing apparatus employing an FMCW waveform having a waveform

465 repetition interval and bandwidth related such that the product of the waveform

repetition interval and the bandwidth is an integer.

2. Signal processing apparatus as claimed in claim 1 in which the waveform comprises a

plurality of identical sweeps, each sweep having a waveform repetition interval and

470 bandwidth such that the product of the waveform repetition interval and the bandwidth

is an integer.

3. Signal processing apparatus as claimed in Claim 2 in which the waveform comprises a series of dwells, each comprising a plurality of identical sweeps.

475

4. Signal processing apparatus as claimed in claim 3 in which a window function is applied at the start and finish of each dwell.

5. Signal processing app °.ratus as claimed in claim 4 in which the window function is a 480 Tukey window function.

6. Signal processing apparatus as claimed in any preceding claim in which the frequency spectrum of the waveform is weighted.

485 7. Signal processing apparatus as claimed in claim 6 in which the waveform frequency spectrum is weighted by applying a window function to it.

8. Signal processing apparatus as claimed in claim 7 in which the window function used

is a cos^πx- Tukey window function.

490 9. Signal processing apparatus according to claim 7 or 8 in which the window function

comprises a cos"x function, in which n is non zero and positive convolved with an extending function having a constant non-zero value across a single continuous range

and a value of zero elsewhere.

495 10. Signal processing apparatus as claimed in claim 9 in which n is 1.

11. Signal processing apparatus as claimed in claim 9 or 10 where the window function Wz is defined by the equation;

W_z = -L . ∑ G" E_z__χ ^■z~--(x+y)_..._(x+y) g a--X ^x

500 where

G = cos" — : x = -X,..._X n≥O 1 +Jfc

Ey ^■y -Y,..,Y

12. Signal processing apparatus as claimed in claim 9 or claim 10 in which the square root of the result of the convolution is used as the window function.

505 13. Signal processing apparatus as claimed in claim 12 where the window function Wz is

defined by the equation;

. X n

W. = — . ∑ G E_z__χ -z~--(x+y),..._(x+y) a=-X where

G /-< = cos n ^xπ x = -X,...X n≥O

1+Jt

_Ey = 1 :y = -Y,...,Y

1 (X+Y

Cg = jz ∑ W_z

14. Signal processing apparatus as claimed ir. any υf claims 9 to 13 which employs a 510 waveform spectrally weighted by the application of a window function in which the value of the index n is selected to minimise the peak to mean ratio of the waveform.

15. Signal processing apparatus as claimed in claim 14 in which the value of the index n is

selected to make the first and second Fresnel peaks have equal magnitude.

515

16. A method of generating an FMCW waveform by defining the waveform repetition interval (T) and bandwidth (B) such that T.B. = an integer.

17. A method of generating a waveform as claimed in claim 16 in which the waveform is

520 generated by evaluating the equation.

g(t) = (\-jTB)- ∑ G_π.e *-* ⁺ "^{■ *}

where M = (TB-l)/2

M

G_=\ π=-M

18. A method of generating a waveform as claimed in claim 19 in which

a = WTB

b = 2U(t/T)

and

TT TB->\ c = — [-1 + 2TB + 1] for phase continuity (TB odd case only)

4 2

25

19. A signal processing apparatus employing a waveform generated by the method of any one of claims 16 to 18.

20. A signal processing method employing a waveform generated by the method of any one of claims 16 to 18.

530

21. A signal processing method employing a waveform having a waveform repetition

interval and bandwidth related such that the product of the waveform repetition interval

and the bandwidth is an integer.

535 22. A signal processing method as claimed in claim 21 in which the waveform comprises

a plurality of identical sweeps, each sweep having a waveform repetition interval and

bandwidth such that the product of the waveform repetition interval and the bandwidth is an integer.

540 23. A signal processing method as claimed in claim 22 in which the waveform comprises a

series of dwells each comprising a plurality of identical sweeps.

24. A signal processing method as claimed in claim 23 in which a window function is applied at the start and finish of each dwell.

545

25. A signal processing method as claimed in claim 24 in which the window function is a

Tukey window function.

26. A signal processing method as claimed in any one of claims 21 to 25 claim in which the 550 frequency spectrum of the waveform is weighted.

27. A signal processing method as claimed in claim 26 in which the waveform frequency

spectrum is weighted by applying a window function to it.

555 28. A signal processing method as claimed in claim 27 in which the frequency spectrum is

weighted by a Tukey window function.

29. A signal processing method according to claim 27 or 28 employing a window function which comprises a cos"x function, in which n is non zero and positive, convolved with

560 an extending function having a constant non zero value accross a single continuous range

and a value of zero elsewhere.

30. A signal processing method according to claim 29 in which n is 1.

565 31. A signal processing method according to claim 29 or 30 where the window function w_z

W_z = -±- . ∑ G E_z__χ x~--(x₊y)_....,(x+y)

<->g a—X ^x

-i n XK G = COS x = -X,...JC n≥O

1 +Jfc

Ey = 1 :y = -Y,...,Y

. (X+T) N z=-(x+τ) 570

32. A signal processing method as claimed in claim 29 or 30 in which the square root of the

result of the convolution is used as the window function.

33. A signal processing method as claimed in claim 32 where the window function w₂ is

575 defined by the equation;

W. ^~- cr_g ^■ Σ ^G" £.-, ^χ-- ^χ+y),...J(^χ+y) a=-X

Ey = 1 :y = -Y,...,Y

34. A signal processing method as claimed in claim 31 or 33 which employs a waveform spectrally weighted by the application of a window function in which the value of the

index n is selected to minimise the peak to mean ratio of the waveform.

580

35. A signal processing method as claimed in claim 34 in which the value of the index n is selected to make the first and second Fresnel peaks have equal magnitude.