Page 24 - Radar Technology Encyclopedia

14 ambiguity functions for typical waveforms ambiguity functions for typical waveforms

Table A1
Ambiguity Functions of Common Waveforms

RMS Widths in
Correlation Widths
Waveform Ambiguity Function Time and Resolution Factors
(at 0.75 level)
Frequency
f
u(t) |c (t, )| a b t f dc 1/B e 1/t e
c
d d
1. Gaussian pulse
2 1.15 1
æ pt 2 ö æ æ t d 2 2 ö ö 2pt p 2 2 ´ 1.15 ---------------- 2t e ------------
¤
-------------------t
e
exp – ç -------- ÷ exp ( j2pf t ) exp – ç 2p -------- + t f ÷ ÷ -------------- p e 2pt e 2t e
ç
e d
d d
t
2
2
è 2t ø è è 4t ø ø e
e e
2. Rectangular pulse
sin ( p f t –( t )) 2
exp ( j2pf t ) t £, t 2 æ d d d ö 3p pt 1 2t 1
¤
0 -------------------------------------------- , t £ t ------ ------- ( 2 – 3)t) ---------- ----- ---
è pf t ø d 2t 3 t
d 3 3t
0, |t | > t 0 /2
0, |t | > t /2
0
3. LFM Gaussian pulse
2 2
2 æ ( æ 1 + 4K ) t
æ pt ö ç w d
ç d ÷ æ æ Bt öö exp ç ----------------------------------
exp – --------- exp j2p f + ------ t ç 2 – p ç 2
ç 2 ÷ è è 0 2t øø è 4t 1.15 1.15 1 1
è 2t ø è e 2p B 2pt -------------- ---------------- ----------- ------------
e e
2 2 ö ö 2pB 2pt e 2B 2t e
+ t f + 2K f t ÷
e d w d d ÷ ÷
ø ø
4. LFM rectangular pulse
2
æ æ K t t ö ö
æ æ Bt ö ö æ w d ö æ d ö
exp j2p f + ------ , t £ t 2 ¤ ç sin ç p ------------- + f t 1 + ------- ÷ ÷
d ø è
è è 0 2t ø ø ç è è t t ø ø ÷ pt 1 1 1 1
ç -------------------------------------------------------------------------------- ÷ 2p B ------- ----------- ---------- --- ---
ç K t ÷ 3 p B 3t B t
w d
æ
ç p ----------- + f t ö ÷
è è t d ø ø
0 t >, t 2 ¤
5. PSK pulse
m – 1 sin ( 2 t
pf t ) ö
å ut it t ,( – r r ) g exp ( j2pf t ) æ ------------------------ ÷ 1 – ------- , t £ t 1 1
d
æ
d ö
ç
0
i
3p
d
-------
i = 0 è pf t ø è t ø d n -------- pt ( 2 – 3)t) ---------- 7t --- t
-----
2t 3 3t 6
n
t £ t 2 ¤ sin ( pf t) 2
1 æ d ö
---- ------------------------ ÷ t > t
,
ç
N è pf t ø d n
0, t > t 2 ¤ d
6. Pulse train
As for As for As for As for
n ( 1 B )
¤
n å c t – it f , ) 2 single- enve- single- enve- --------------------- t æö æö
1
1
e p
´
----
(
----
å A () a ( tit t , ) p d r d t r t èø èø
t
t
–
pt
r r
pt
i = – n pulse lope pulse lope A pt r e p e a
i = 0
¥ 2 a pt A pt a pt ´ æ ------
1 ö
(
´ exp ( j2pf t ) t £, t ¤ 2 ´ å c t , f – kt ¤ ) è B ø
0 0 a d d r e a
k = – ¥
0, |t | > t 0 /2
Note: g is a series of unities, e.g., (1, - 1, - 1, ... ) defining the binary code; t is the subpulse width; A is the pulse-train envelope; a is the single-pulse
pt
n
i
pt
2
2
modulation function; t is the pulse repetition interval; t is the transmitted pulse width; |c (t , f )| is the single-pulse ambiguity function; and |c (t , f )| is
r
p d
a d
d
d
the pulse-train envelope ambiguity function.