WO1995032437A1 - Signal processing apparatus and method - Google Patents
Signal processing apparatus and method Download PDFInfo
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- WO1995032437A1 WO1995032437A1 PCT/GB1995/001163 GB9501163W WO9532437A1 WO 1995032437 A1 WO1995032437 A1 WO 1995032437A1 GB 9501163 W GB9501163 W GB 9501163W WO 9532437 A1 WO9532437 A1 WO 9532437A1
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- Prior art keywords
- waveform
- signal processing
- window function
- processing apparatus
- processing method
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Classifications
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- G—PHYSICS
- G01—MEASURING; TESTING
- G01S—RADIO DIRECTION-FINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCE-DETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES
- G01S13/00—Systems using the reflection or reradiation of radio waves, e.g. radar systems; Analogous systems using reflection or reradiation of waves whose nature or wavelength is irrelevant or unspecified
- G01S13/02—Systems using reflection of radio waves, e.g. primary radar systems; Analogous systems
- G01S13/06—Systems determining position data of a target
- G01S13/08—Systems for measuring distance only
- G01S13/32—Systems for measuring distance only using transmission of continuous waves, whether amplitude-, frequency-, or phase-modulated, or unmodulated
- G01S13/34—Systems for measuring distance only using transmission of continuous waves, whether amplitude-, frequency-, or phase-modulated, or unmodulated using transmission of continuous, frequency-modulated waves while heterodyning the received signal, or a signal derived therefrom, with a locally-generated signal related to the contemporaneously transmitted signal
Definitions
- This invention relates to a signal processing apparatus and method and in particular to
- FMCW continuous wave
- the invention enables a waveform, for use in a radar system for example, to be generated
- 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.
- 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.
- 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.
- 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
- This invention provides a signal processing method and apparatus for producing and
- 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.
- 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.
- window function is a Tukey window function.
- the frequency spectrum of the waveform can be weighted, preferably by applying a window function such a Tukey window function to it.
- a window function such as a Tukey window function
- this invention provides a window function comprising a cos"x
- a window function Wz can be used defined by the equation
- the window function is the square root of the result of the convolution.
- a window function Wz can be used defined by the equation; 5
- 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.
- T waveform repetition interval
- B bandwidth
- the waveform is generated by then evaluating the equation
- variables a, b and c can be defined as;
- 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.
- 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
- a window function can be applied at the start and finish of each dwell
- a suitable window function is a Tukey window function.
- the frequency spectrum of the waveform can be weighted, preferably
- Figure 1 shows the frequency spectrum of a typical conventionally generated waveform
- Figure 2 shows the frequency spectrum of a waveform produced employing the
- 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
- FIGS. 4A to 4D are explanatory diagrams showing Tukey dwell weighting functions
- Figures 4E to 4H respectively show the waveform spectral occupancies produced by the
- Figures 5A to 5D are explanatory diagrams showing Tukey spectral weighting functions having different values of taper
- 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
- 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
- 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
- Figures 9A to 9D show the cos ⁇ Tukey spectral weighting function with 20% taper for
- FIGS 9E to 9H respectively show the waveform envelope magnitude functions
- Figures 10A to 10D show the square root cosTukey spectral weighting function for 20% taper different values of n;
- 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 cos n Tukey spectral related waveform
- 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
- Figure 14 shows signal processing apparatus for use in a radar system employing the
- a (t) is some amplitude scaling factor (usually 1).
- LFM linear frequency modulated waveform
- a radar or sonar often uses a waveform termed a "dwell” and comprising a number of sweeps.
- the dwell waveform g"(t) (also time limited) is obtained by replicating the time-limited signal g'(t) at intervals of T
- Ns Number of sweeps in a dwell
- 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
- the waveform is continuous and as a result the frequency spectrum
- T x B integer
- 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
- 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.
- the waveform can be exactly represented by digital samples and the waveform can be precisely
- T.B 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
- the basic waveform is obtained in the time-domain by means of a inverse finite Fourier
- M g(t) ( ⁇ -jTB). ⁇ G ⁇ .e i - an2 + *" + c >
- ⁇ TB* ⁇ c — [-1 + 2TB + 1] for phase continuity (TB odd case only)
- 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.
- 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
- 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
- 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.
- 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.
- FIG. 1 and 2 illustrate a typical spectral template for out-of-band emission levels to allow simple comparison between the conventional and inventive waveforms.
- 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.
- the decay rate is .18dB/octave increasing by .6db/octave for each doubling of
- 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
- 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.
- 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.
- Figures 8A to 8H illustrate the same scenario as in Figures 7 for the cos n -Tukey window with 50% taper.
- Figures 9A to 9D show the cos"Tukey spectral weighting function with 20% taper for
- 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
- Figures 10A to 10D show the square root cos" Tukey spectral weighting
- 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.
- 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.
- Minimum peak-to-mean i.e. ⁇ 0.9dB
- Figure 14 shows signal processing apparatus for use in a radar system employing
- the waveforms are produced by
- the r.f. signal from the upconverter 2 is supplied to a high power amplifier 3 which is connected to an antenna 4 for
- 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.
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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 Ez__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) = Gn.e *■«* * bn -i* c)
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 cosπTukey 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). ∑ Gπ.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)
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 ∑ Wk
N *=0
Figures 3A to 3D illustrate the effect of application of a 3,125% taper Tukey window on a conventional waveform for Ns = 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 Ns = 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 cosπx- 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;
Wz = -L . ∑ G" Ez_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 Ez_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 ∑ Wz
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)- ∑ Gπ.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 wz
Wz = -±- . ∑ G Ez_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. ~- crg ■ 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.
Priority Applications (1)
Application Number | Priority Date | Filing Date | Title |
---|---|---|---|
AU25321/95A AU2532195A (en) | 1994-05-20 | 1995-05-22 | Signal processing apparatus and method |
Applications Claiming Priority (4)
Application Number | Priority Date | Filing Date | Title |
---|---|---|---|
GB9410164A GB2290188A (en) | 1994-05-20 | 1994-05-20 | Radar range processing |
GB9410164.9 | 1994-05-20 | ||
GB9421466A GB2291301A (en) | 1994-05-20 | 1994-10-24 | Signal processing apparatus and method |
GB9421466.5 | 1994-10-24 |
Publications (1)
Publication Number | Publication Date |
---|---|
WO1995032437A1 true WO1995032437A1 (en) | 1995-11-30 |
Family
ID=26304917
Family Applications (1)
Application Number | Title | Priority Date | Filing Date |
---|---|---|---|
PCT/GB1995/001163 WO1995032437A1 (en) | 1994-05-20 | 1995-05-22 | Signal processing apparatus and method |
Country Status (2)
Country | Link |
---|---|
AU (1) | AU2532195A (en) |
WO (1) | WO1995032437A1 (en) |
Citations (1)
Publication number | Priority date | Publication date | Assignee | Title |
---|---|---|---|---|
EP0425006A1 (en) * | 1989-10-24 | 1991-05-02 | Hollandse Signaalapparaten B.V. | FM-CW radar apparatus |
-
1995
- 1995-05-22 WO PCT/GB1995/001163 patent/WO1995032437A1/en active Search and Examination
- 1995-05-22 AU AU25321/95A patent/AU2532195A/en not_active Abandoned
Patent Citations (1)
Publication number | Priority date | Publication date | Assignee | Title |
---|---|---|---|---|
EP0425006A1 (en) * | 1989-10-24 | 1991-05-02 | Hollandse Signaalapparaten B.V. | FM-CW radar apparatus |
Non-Patent Citations (1)
Title |
---|
YVES M. DE VILLERS: "A coherent Ka band FMCW radar for real time target acquisition", THE RECORD OF THE 1993 IEEE NATIONAL RADAR CONFERENCE, 20 April 1993 (1993-04-20), LYNNFIELD , MA , USA, pages 1 - 5, XP000389736 * |
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