3.0 WAVEFOM CODING 3.1 Introduction We now want to loo at waveform coding. We secifically want to loo at hase and frequency coding. Our first exosure to waveform coding was our study of LFM ulses. In that case the frequency and hase changed continuously across the ulse. In the cases we consider here, the frequency or hase changes in discrete stes across the ulse. As with LFM, the amlitude is constant across the ulse. In our studies we will subdivide the ulse into a series of subulses, or chis, where the duration of the chis is the same. This is not a requirement but a convenience for our uroses, and in most ractical alications. We then assign a different hase and/or frequency to each chi according to some rule defined by the frequency or hase coding algorithm. We assume that the amlitude of all chis is the same. This is also a semi requirement of most hase and frequency coding schemes in that they were develoed under the assumtion that the amlitudes of the chis are the same. In most alications, the chis are adjacent. That is, the waveform has a 100% duty cycle. Again, this is not a hard requirement but is a standard to which waveform designers adhere. An excetion to this is the ste frequency waveform. In this case the sacing between ulses is usually large enough to ermit unambiguous range oeration, although this is not a hard requirement. An examle of a 7-chi, Barer coded waveform is shown in Figure 1. This figure can be considered an ideal, base-band signal in that the ulses are erfectly rectangular and the voltage values are normalized values of either +1 or -1. Figure 1 7-chi Barer Coded Waveform We usually resort to waveform coding when we simultaneously desire fine range resolution and high target energy. The desire for fine range resolution dictates a narrow ulse while the desire for high energy dictates a wide ulse. With roerly chosen hase or frequency coding we can usually achieve both desires The most common form of waveform coding is LFM. LFM waveforms are reasonably easy to generate and rocess. They also lend themselves to reasonably straight forward analysis. In site of this, researchers have studied, and still study, many other waveform coding methods. Interestingly, several of these are based on the LFM waveform. Examles include frequency ste waveforms and Fran olyhase coded waveforms. 2014 M. C. Budge - merv@thebudges.com 1
In here we will resent several different waveform coding schemes and discuss the characteristics of their ambiguity functions. To that end, I ve sent you a Matlab scrit that generates the ambiguity function. Some of you may already have such a rogram and, if so, I encourage you to use your routine. 3.2 Fran Polyhase Waveforms We will start by examining a digital form of LFM that is termed Fran olyhase coding. In essence, a Fran olyhase code is a digital reresentation of a quadratic hase shift, the hase shift exhibited by LFM. 2 Fran olyhase codes have lengths that are erfect squares; i.e. N L where L is an integer. The code can be formed by first creating an L L matrix of the form F L 0 0 0 0 0 0 1 2 3 L 1 0 2 4 6 2L 1. (1) 0 3 6 9 3L 1 0 L 1 2L 1 3L 1 L 1 2 Next the rows or columns are concatenated to form a vector of length N Finally, the hase is determined by multilying each element by 2 L. 2 L. (2) We will illustrate this by an examle. We consider L 4 which roduces an 2 N L 16 element Fran olyhase code. The Fran olyhase matrix is 0 0 0 0 0 1 2 3 F4 0 2 4 6 0 3 6 9 and 2 4 2. (4) The vector of hase shifts is m 0 0 0 0 0 1 2 3 0 2 4 6 0 3 6 9 2. (5) The resulting Fran olyhase coded waveform is N 1 j m t m u t e rect FP. (6) m0 The range resolution of ufp t is. 2014 M. C. Budge - merv@thebudges.com 2 (3)
The ambiguity function of a ufp t with the 16 element Fran olyhase code of Equation (5) is shown in Figure 2. In the lot of Figure 2, Doler ranges from 0 to 1 and range delay goes from 16 to 16 where 16 is the total duration of the ulse. Figure 2 Ambiguity function of a 16 Chi Fran Polyhase Waveform The deiction of the ambiguity function shown in Figure 2 allows us to visualize the structure of the overall ambiguity function while still being able to, f, the ambiguity visualize the matched-doler range cut (the lot of function, vs. for f 0 and hereinafter termed the range cut). It will be noted that the ambiguity function of the Fran olyhase waveform exhibits some semblance of the ridge characteristic of LFM waveforms. We might have exected this since the Fran olyhase waveform is a sort of discrete version of the LFM waveform. Figure 3 contains a lot of m (with aroriate hase unwraing) for the 16 chi examle above. It also contains a lot of the hase shift of an LFM waveform that has a BT roduct of 16, the same as the BT roduct of the 16 chi Fran olyhase waveform. As can be seen, the Fran olyhase waveform has aroximately the same quadratic hase characteristic as an equivalent LFM waveform. 2014 M. C. Budge - merv@thebudges.com 3
Figure 3 Phases for 16-chi Fran Polyhase and Equivalent LFM 3.3 Zadoff-Chu Waveform There are several variants on Fran olyhase (see eference 1 1 ) coding. One that is almost directly derived from LFM is termed a minimal chir sloe Zadoff-Chu code. This code can be any length, N. The hase shifts are given by 2 m N even N m m 0,1, N 1. (7) m 1 m N odd N Figure 4 contains lots of m for a 16 chi Fran olyhase, a 16 chi Zadoff-Chu coded waveform and an LFM waveform with a BT roduct of 16. As can be seen, the hase shift of the Zadoff-Chu waveform more closely aroximates the hase shift of the LFM waveform than does the hase shift of the Fran olyhase waveform 1 Levanon, N and Mozeson, E, adar Signals, John Wiley and Sons, Hoboen, N.J., 2004, ISBN 0-471-47378-2 2014 M. C. Budge - merv@thebudges.com 4
Figure 4 Phases for 16-chi Fran Polyhase and Zadoff-Chu, and an Equivalent LFM Fran olyhase and Zadoff-Chu codes are called olyhase codes because the hase shifts of the various chis can tae on a multitude of values. This maes their generation slightly comlicated in that they require a quadratic modulator to obtain the hases. They also require that the signal rocessor have the ability to rocess comlex signals. With modern hardware this is not an issue. In older hardware it osed roblems. 3.4 Barer Coded Waveforms A simlification of olyhase coded waveforms are waveforms that use only two hase shifts of 0 and. These are termed binary hase codes. In older hardware these were easy to generate and rocess since they only involved sign reversal. A common set of binary hase codes found in radar texts are the Barer codes. These codes have the interesting roerty that the ea sidelobe level of the range cut (the range sidelobes) is 1 N. (This assumes that the ea of the ambiguity function is normalized to unity.) Although Barer codes have low range sidelobes, their sidelobe levels off of matched Doler can be high as shown in Figure 5. 2014 M. C. Budge - merv@thebudges.com 5
Figure 5 Ambiguity Function of a 11-chi Barer Coded Waveform There only 7 nown Barer codes. They have lengths of 2, 3, 4, 5, 7, 11 and 13. The hase shifts for the 7 codes are shown in Table 1. Code Length Table 1 Phase Shifts for Barer Codes Phase Shifts 2 0 0 or 0 3 0 0 4 0 0 0 or 0 0 0 5 0 0 0 0 7 0 0 0 0 11 0 0 0 0 0 13 0 0 0 0 0 0 0 0 0 2014 M. C. Budge - merv@thebudges.com 6
The reference indicated in footnote 1 contains a tabulation of olyhase Barer codes that range in length from 4 to 45. As imlied by their name, these waveforms use olyhase coding and exhibit ea sidelobe levels of about 1 N. 3.5 PN Coded Waveforms Another tye of binary hase coded waveforms is seudo-random-noise, or PN, waveforms. They derive their name from the fact that the algorithm used to generate the 0 and hase shifts is also used to generate seudo random numbers. PN waveforms are also used in sread sectrum communications and to scramble (or whiten) signals in HDTV and other digital transmission systems. M PN waveforms most often have lengths of N 2 1 where M is an integer. The sequence of hase shifts is determined by a feedbac shift register device similar to that shown in Figure 6. In that figure the boxes reresent binary shift register elements (fli-flos) and the adder is a modulo-2 adder. The characteristics of the sequence of ones and zeros roduced by the feedbac shift register are determined by which shift register stages are summed and fed bac to the left-most shift register stage. Ideally, we want to choose the stages M so that the sequence of ones and zeros reeats only after N 2 1 samles, where M is the number of stages in the shift register. Such a sequence of ones and zeros is termed a maximal length PN sequence. Solni 2 has, on age 10.20, a tabulation of feedbac configurations needed to generate maximal length sequences with lengths u to 2 20-1. The table contains only one feedbac configuration for each code length. However, it indicates that there are generally several to many feedbac configurations for each length. There are texts and web-sites that rovide other feedbac configurations. When searching for these you might use eywords such as maximal length sequences shift register sequences, linear shift register sequences and seudo random noise. To generate a PN sequence, one must initially load the shift register with any binary number excet zero. Figure 6 M-stage Feedbac Shift egister 2 Solni, M. I., adar Handboo Second Edition, McGraw-Hill, New Yor, N.Y., 1990. 2014 M. C. Budge - merv@thebudges.com 7
Figure 7 contains a lot of the ambiguity function for a 15-chi PN waveform where the hase coding was generated with the feedbac configuration from your text and an initial load of 0001. It will be noted that the range cut doesn t have sidelobes that are as low as for the Fran olyhase or Barer waverorms. However, the ambiguity function doesn t have the large eas that the other two waveforms exhibit. This is a characteristic of PN coded waveforms: their sidelobe levels are generally oay but not extremely low or high. Long PN coded waveforms have ambiguity functions that aroach the ideal, thumbtac ambiguity function. Figure 7 Ambiguity Function of a 15-chi PN Coded Waveform We noted above that we generated the hase code by using an initial load of 0001. It turns out that the initial load can have a fairly significant imact on the matched-doler range sidelobes of the ambiguity function. It also has a lesser imact on the other range-doler sidelobes. The only nown way to choose an initial load that rovides the desired sidelobe characteristics is to exeriment. As a side note: changing the initial load results in a hase coding sequence that is a cyclic shift of the original sequence. This is a roerty of maximal length PN codes: they are eriodic with a eriod of 2 M -1. 2014 M. C. Budge - merv@thebudges.com 8
3.6 Mismatched PN Processing We now want to investigate a secial tye of rocessing of PN coded waveforms that taes advantage of an interesting roerty of PN codes. The roerty we refer to is that the circular autocorrelation of a PN coded sequence has a value of either N or -1. With a circular correlation, when we shift the sequence to the right by n chis we tae the n chs that fall off of the end of the shifted sequence and lace them at the beginning of the shifted sequence. This is illustrated in Figure 8. In this figure we have used the 7-bi PN code of 1001110 to generate the PN coded sequence of -111-1-1-11. Figure 8 Illustration of a 2-bit Circular Shift To erform the circular autocorrelation we mae a coy of the sequence to roduce two sequences. We then circularly shift one sequence by n chis, multily the result in N chi airs and form a sum across the N chi result. This is illustrated in Figure 9. Mathematically we can write the circular correlation as N 1 rn r n (8) n0 N N where m denotes evaluation of m modulo N. The interesting roerty of PN coded sequences referenced above is that N 0, N,2 N, 1 otherwise. (9) 2014 M. C. Budge - merv@thebudges.com 9
Figure 9 Illustration of Circular Convolution for =2. We now want to aly this roerty to examine a secial tye of PN coded waveforms. We will assume that we encode a 0 of the PN code to a hase of 0 and a 1 of the PN code to a hase shift of. Thus a single PN coded ulse corresonding to the 7-bit PN code of Figures 8 and 9 would be as shown in Figure 10. Figure 10 7-chi, PN Coded Waveform We assume that the transmit waveform, We define a matched filter that is matched to a signal concatenation of three ut s. Thus ut, is as shown in Figure 10. vt where vt is a vt would be as shown in Figure 11. The t 0 reference oints in Figures 10 and 11 are used to denote the time alignment for matched range. Thus, when the received signal (a scaled version of Figure 10) is aligned with the center of the three PN coded ulses of Figure 11 the matched filter is matched in range to the received ulse. Figure 11 Waveform to which Matched Filter is Matched 2014 M. C. Budge - merv@thebudges.com 10
The matched filter outut, which is also the range cut of the cross ambiguity function of u t and v t, is j2 ft vmf uth tdt, f u tv t e dt f 0 f 0. (10) We want to examine MF integer between N 1 of utv t n PN coded waveform and n 2. v for n where is the chi width and n is an and N 1. We articularly want to examine the form. This is illustrated in Figure 12 for the aforementioned 7-chi When we form utv t 2 we get N 1 j j 2 t N utv t 2 e e rect (11) 0 and t v e e dt e 1 1 2 2 N N j j j N N MF 2 rect. (12) 0 0 We note from Figure 12 that is equal to either 0, or. We note 2 N further that there are 3 cases where 2 0, 2 cases where and 2 cases where. With this we get 0 2 N j j j v 2 3e 2e 2e 3 4cos. (13) MF It turns out that for all N 1 Figure 12 Formation of utv t 2 N 2 N 2014 M. C. Budge - merv@thebudges.com 11
MF 3 4cos v. (14) In fact, for any N-chi (N=2 M -1) PN coded waveform with by the above rule, v N ut and vt chosen N 1 N 1 cos 1. (15) 2 2 MF Stated in words, the range sidelobes within N-1 chis of the mainlobe have a constant value as given by Equation (15). As an interesting extension of the above, if we choose such that N 1 N 1 cos 0 (16) 2 2 or 1 1 N cos 1 N we get v 0 N 1 MF (17). (18) In words, the range sidelobes within N-1 chis of the mainlobe are zero. This has the otential of being useful in the situation a radar must be able to detect a very small target in the resence of a very large target, rovided both targets are at the same Doler. Figure 13 contains a lot of the ambiguity function for the 7-chi PN examle above. In the case the hase,, was chosen to be 1 7 1 7 1 1 cos cos 0.75 138.59. (19) 2014 M. C. Budge - merv@thebudges.com 12
Figure 13 Ambiguity Function for 7-chi Examle Above It will be noted that the range sidelobes around the central ea are, indeed, zero. However, it will be noted that the sidelobes off of matched Doler rise significantly. It will also be noted that the range cut contains two extra eas. These eas are termed range ambiguities and are due to the fact that u t correlates with each of the other two PN coded ulses that mae-u v t. The range sidelobes adjacent to these range ambiguities are the normal range sidelobes associated with PN coded waveforms. 3.7 Comlementary Coded Pulses The reference in footnote 1 contains a discussion of comlementary ulses, or waveforms using comlementary coded chis. These waveforms are not PN-based but also exhibit zero range sidelobes around the central ea. A binary, comlementary coded waveform is illustrated in Figure 14 and its ambiguity function is illustrated in Figure 15. As noted with the PN-coded waveform, the range sidelobes around the central ea are zero. However, the sidelobes off of matched Doler are fairly large. The region of zero sidelobe level can be extended by extending the duration of the zero art of the waveform of Figure 14. 2014 M. C. Budge - merv@thebudges.com 13
Figure 14 Examle of a Comlementary Coded Waveform Comlementary coded waveforms derive their name from the fact that the comonents that maeu the waveform have range sidelobe levels that are negatives of each other. For examle, the range sidelobes of the left-hand, fourulse waveform of Figure 14 are the negative of the sidelobes of the right-hand four-ulse waveform. The fact that the waveforms are combined into a single waveform, with a dead sace equal to at least the duration of each waveform, causes the sidelobes of one to add to the sidelobes of the other. Since the sidelobes have oosite signs, their sum is zero. Figure 15 Ambiguity Function of the Comlementary Coded Waveform of Figure 14 2014 M. C. Budge - merv@thebudges.com 14
3.8 Generation and Processing of Phase Coded Waveforms To conclude our discussion of hase coded waveforms, we want to discuss, at least concetually, how such waveforms are generated and how a matched filter might be imlemented. Figure 16 contains a bloc diagram of a method that might be used to generate a hase coded waveform with arbitrary hase shifts on each ulse. The IF Oscillator generates quadrature signals at some intermediate frequency. The outut of the synchronizer consists of two sequences of ulses where one sequence has amlitudes that deend uon the cosine of the hases on the individual ulses and the other sequence has amlitudes that deend uon the sine of the hases on the individual ulses. For the case of the comlimentary coded waveform of Figure 14, the cos would aear as shown in Figure 14 and the sin outut would be zero. If the hases of the were other than 0 and the sin outut would be non-zero. The left two mixers of Figure 16 are termed baseband mixers and, for all ractical uroses, are multiliers. The outut of the to mixer is cos cos and the outut of the bottom mixer is sin sin. The outut of the summer is cos cos t sin sin t cos t. (20) IF IF IF As can be seen from Equation (20), the combination of the IF oscillator, synchronizer, two mixers and summer roduce an IF signal that has the roer hase modulation imosed uon it. IF t IF t Figure 16 Signal Generator for Phase Coded Waveforms The right mixer is a single-sideband mixer that roduces the carrier signal with the roer hase modulation. 2014 M. C. Budge - merv@thebudges.com 15
A bloc diagram of a concetual matched filter for a hase coded ulse is shown in Figure 17. The mixer in the uer left heterodynes the received signal, with the hase modulation, to some IF. The IF signal is then rocessed by a matched filter matched to a single chi. The outut of the matched filter is then converted to digital data via a comlex ADC. The ADC samles once er chi. That is, the sacing between samles is. The samles are then further comressed by the ulse comressor. Figure 17 Concetual Matched Filter for a Phase Coded Waveform The ulse comressor first converts the digital word to a comlex number and feeds it to an N-1 stage digital delay line (a multi-bit, N-1 stage shift register). The oututs of the various stages of the digital delay line are multilied by comlex exonentials with negative hase shifts. It will be noted that the order of the hase shifts is reversed. This, in effect, is an v t. Finally, the oututs multiliers are summed to form imlementation of the comressed outut. It will be noted that the ulse comressor is a correlator (or convolver). The ulse comressor of Figure 17 is designed as an all-range rocessor. That is, it continually oerates over the entire waveform PI. Because of this, the multilies and the sum must all be comleted within a single chi duration, 2014 M. C. Budge - merv@thebudges.com 16
or. Further, the multiliers and the summer are comlex oerators. Given this, very high seed digital hardware is needed to rocess hase coded waveforms of even moderate bandwidth. The rocessing load can be relieved if the waveform is used over only a limited range window. In this case, the correlator in the bottom art of Figure 17 would be relaced with a combination of an FFT, a comlex, vector multilier, and an inverse FFT. The length of the FFT, multilier and inverse FFT would deend uon the size of the range window, in chis and the number of chis in the hase coded waveform. Secifically, if the range window is N chis long and the hases coded waveform contains N chis, or subulses, then the minimum FFT length would be N 2N N. (21) FFT W The factor of 2N is needed to account for the fact that the outut of the matched filter is two times longer than the overall ulse width. As an examle, suose one wanted to rocess a range window of 15 Km and was using a hase coded waveform with a bandwidth of 10 MHz and a duration of 10 µs. The 10 MHz bandwidth translates to a chi width of 0.1 µs, or 15 m. With this we get N 15000 15 1000 (22) W and 10 s N 100. (23) 0.1s From these two values we get N 2N N 2 100 1000 1200. (24) FFT W This means that the ulse comressor will require a 2048 oint FFT and IFFT, and will require 2048 comlex multilies. The combination of the FFT, IFFT and multily would need to be comleted in one PI. If one uses a PI of say, 500 µs, then the oerations would need to be comleted in 500 µs. This is still stressing but may be easier than the all-range imlementation of Figure 17. In some alications where latency is not a major issue, the FFT, multily and IFFT oerations can be ielined to ease comuting requirements. In such an alication the FFT would be erformed on one PI, the multily on the next PI and the IFFT on the third PI. Further, while the IFFT is being erformed on the K th ulse, the multily would be erformed on the (K+1) th ulse and the FFT would be erformed on the (K+2) th ulse. If the hase coding is restricted to binary, with hase shifts of 0 and, rather than olyhase, the generation and rocessing of the hase coded waveforms becomes considerably simler. In Figure 16 the sine channel can be eliminated and in the ulse comressor the arithmetic becomes real instead of comlex. In some instances the arithmetic is further simlified to single-bit arithmetic. 2014 M. C. Budge - merv@thebudges.com 17 W
3.9 Ste Frequency Waveforms 3.9.1 The Basics We now want to discuss some of the math behind, and roerties of, ste frequency waveforms. The transmit signal we are interested in is shown in Figure 18. For this analysis, and in most ractical alications, we assume that the radar oerates unambiguously in range. That is, the signal from ulse is received before ulse +1 is transmitted. Thus, we can thin of rocessing one ulse at a time and saving the results for later, further, rocessing. This is similar to the aroach we used when we studied SA. Figure 18 Ste Frequency Waveform o We assume that the frequency, f, of the th ulse is given by f f f (25) where f o is the carrier frequency and f is the frequency ste. For now we assume that the individual ulses are unmodulated. With this we can write the normalized transmit signal for the th ulse as t. (26) 2 o v j f f t T t e rect In this equation, t 0 for each ulse is referenced to the center of the ulse. This is imortant in terms of imlementation because it carries the tacit assumtion that the signal generator that creates the transmit and heterodyne signals are erfectly coherent, i.e. derived from the same source. The normalized signal returned from a target at a range delay of is j2 fo f t t v t e rect. (27) We assume that we now well enough to be able to samle the matched filter outut near its ea. A more accurate measurement of will be obtained from the outut of the ste waveform signal rocessor. For now we assume that the radar and target are fixed so that is constant. 2014 M. C. Budge - merv@thebudges.com 18
The heterodyne signal is given by h t e j2 fo f t. (28) We note that the frequency of the heterodyne signal is different for every ulse and that v t. With this, the outut of the h t is erfectly coherent with heterodyne oeration is j2 fo j2 f rect t vh t v t h t e e. (29) The first term is a constant hase shift that will be common to all ulses. We will lum it into some constant that we will dro. For the next ste we ass vh t rect to roduce a normalized outut of j2 f tri t vm t e where of unity. v M 2014 M. C. Budge - merv@thebudges.com 19 T t through a matched filter matched to (30) tri x is a triangle centered at x 0 with a base width of 2 and a height Finally, we samle 2 e vm t at some, close to, to obtain j f tri. (31) After we obtain vm from N ulses we form the sum V a v a e N1 N1 j2f M tri (32) 0 0 where the a are comlex weight coefficients that we choose to maximize V. We recognize Equation (32) as the form of the sum we have encountered in our antenna, stretch rocessing and SA analyses. We can use this nowledge to ostulate a form of the a as j 2 f a e (33) and write V as N 1 j2 f V tri ae 0 (34)
We can evaluate and normalize Equation 34 to yield V sin Nf tri sin f A lot of V vs.. (35) f is shown in Figure 19 for N 10 and without the tri[x] function. It will be noted that the central ea occurs at f and that the first null occurs at f 1 N 0.1. The other 0 eas, which are range ambiguities, are located at integer values of f. This tells us that the range resolution of the waveform is 1 Nf and that the range ambiguities are located at amb. (37) f (36) Figure 19 Plot of V without tri[x] In the above we ignored the tri[x] function to emhasize the location of range ambiguities. If we now include it, we can quantify the effect of the singleulse matched filter on V. We will add the extra ste of recognizing that V is the matched-doler range cut of the ambiguity function of vt t and, in future references, use V,0 20 contains lots of,0 for 0.5, 1 and 2. The to lot f. With this, Figure 2014 M. C. Budge - merv@thebudges.com 20
corresonds to the case of f 0.5 and the bottom lot corresonds to the case of f 2. The dashed triangles are the single-ulsed matched filter resonses. Figure 20 Plot of,0 for f 0.5, 1 and 2 It will be noted that for f 0.5 and 1 the single-ulse matched filter resonse attenuates the range ambiguities. However, when f 2 the range ambiguities are resent. This interaction of the single-ulse matched filter resonse is a limitation that must be considered when designing ste frequency waveforms. Secifically, if one increases f in an attemt to imrove range resolution one runs the ris of introducing range ambiguities. From this we see that, for a given N there is direct relation between f and the width of the individual ulses. To imrove resolution by increasing f, and avoid range ambiguities, we must assure that f 1. (38) It turns out that a means of effectively reducing is to use a ulse comression waveform on the individual ulses. In that case would be the comressed ulse width. Another means of imroving range resolution would be to increase the number of ulses that are rocessed. However, this must be done with care in that it can have negative consequences in terms of timelines and the otential imact of target motion. 2014 M. C. Budge - merv@thebudges.com 21
3.9.2 Doler Effects We now want to consider the effects of target motion. For now we will be concerned with only target Doler. To include target Doler, we write the target range as t 0 t (39) and the target range delay as 2 t c t f f t. (40) 0 0 d o We can write the target range delay at the time of the th transmit ulse as T f f T (41) 0 d o where T is the PI (see Figure 18). If the transmit signal defined in Equation (26) then the received signal is j2 fo f t t v t e rect. (42) j2 fo f t 0 fd fo T t e rect If we maniulate the above using rect[x] function we get fo f fd fo fd with 0 in the j2 fo f t j2f 0 j2 fdt t 0 v t e e e rect. (43) We note that the aroximation of f f f f f may not be very good. o d o d However, it allows us to focus directly on the effects of target Doler and not be concerned with the otential smearing effects the aroximation could cause. This would need to be considered in a more comlete analysis. If we comare Equation 43 with Equation (27) we note that the only difference is the aearance of the term related to Doler. Thus, if we reeated the heterodyning and matched filtering math from above we would get j2f 0 j2 fdt 0 vm 0 e e tri. (44) If we form a weighted sum of the v M 0 N1 N1 V 0 vm b e 0 0 as we did before we would get j2 f 0 fdt 0 tri. (45) 2014 M. C. Budge - merv@thebudges.com 22
As we did earlier, we want to choose the more general form would be to use b e b to maximize V 0. However, a j2 f ft (46) which would yield N 1 0 j2 0 f f fdt 0, f fd tri e (47) 0 Or, evaluating the sum, sin 0 d N f f f T 0 0, f fd tri sin 0 d Figure 21 contains a lot of f f, 0 f f f T discussed earlier and a PI of 500 µs. We further use d. (48) for the 10-ulse waveform 1 s and f 1 MHz so that f 1. It will be noted that the ambiguity function has the ridge associated with LFM waveforms. This is exected because the ste frequency waveform is a discrete reresentation of a waveform with linear frequency modulation. Figure 21 Ambituity Function of a Ste Frequency Waveform 2014 M. C. Budge - merv@thebudges.com 23
Figure 22 contains a matched-range, Doler cut lot, a lot of 0, f fd vs. f fd, of the ste frequency waveform. It will be noted that the Doler resolution of this waveform is 20 KHz or 1 NT, as exected. Figure 22 Matched-ange, Doler Cut for a Ste Frequency Waveform Figure 23 contains lots of range cuts at matched Doler and at a Doler offset of one Doler resolution cell (200 Hz). It will be noted that a Doler offset of one Doler resolution cell causes a range error of one range resolution cell. This means that the ste frequency waveform is very sensitive to Doler and that, if we want accurate absolute range measurement, the range shift due to target Doler must be removed. This can be done if the target is in trac and the relative velocity between the radar and target is nown with reasonable accuracy. If the ste frequency waveform is used in its more common role of target imaging (or SA), the various scatterers of the target should be moving at about the same range-rate so that range errors due to Doler differences of the scatterers should be small. One would still want to remove the gross Doler so as to minimize losses due to Doler mismatch. (It will be noted from Figure 23 that the range cut at f 1 NT is down about 1 db (-10log(0.9).) 2014 M. C. Budge - merv@thebudges.com 24
Figure 23 ange Cuts of a Ste Frequency Waveform 2014 M. C. Budge - merv@thebudges.com 25