SIGNAL-TO-NOISE RATIO

In this section the results of statistical noise theory will be applied to obtain the signal-to-noise ratio at the output of the IF amplifier necessary to achieve a specified probability of detection without exceeding a specified probability of false alarm. The output signal-to-noise ratio thus ohtained may be substituted into Eq. (2.6) to find the minimum detectable signal, which, in

turn. is used in the radar equation, as in Eq. (2.7).

Consider an IF amplifier with bandwidth

B,F

followed by a second detector and a video amplifier with bandwidth B,. (Fig. 2.3). The second detector and video amplifier are assumed to form an envelope detector, that is, one which rejects the carrier lrequency but passes the modulation envelope. To extract the modulation envelope, the video bandwidth must be wide enough to pass the low-frequency components generated by the second detector, but not so wide as to pass the high-frequency components at or near the intermediate frequency. The video bandwidth B,. must be greater than

B,F/2

in order to pass all the video modulation. Most radar receivers used in conjunction with an operator viewing a CRT display meet this condition and

IF Second Video

amplifier 1 - ~ amplifier

(81,) detector

(Bvl Figure 2.3 Envelope detector.

may be considered envelope detectors. Either a square-law or a linear detector may be assumed since the effect on the detection probability by assuming one instead of the other is usually small.

The noise entering the

IF

filter (the terms filter and amplifier are used interchangeably) is assumed to be gaussian, with probability-density function given by

(2.20) where p(v) dv is the probability of finding the noise voltage v between the values of v and v

+

dv, ,j1₀ is the variance, or mean-square value of the noise voltage, and the mean value of 11 is taken to be zero. If gaussian noise were passed through a narrowband IF filter-one whose bandwidth is small compared with the midfrequency-the probability density of the envelope of the noise voltage output is shown by Rice⁹ to be

(2.21) where R is the amplitude of the envelope of the filter output. Equation (2.2 l} is a form of the Rayleigh probability-density function.

The probability that the envelope of the noise voltage will lie between the values of V₁ and

Vi

is

.Vi R (

Probability (V1

<

R

<

V2 )

= J -:,:-

^exp

V₁ 'f'O

2t/lo

R2)

dR (2.22)

The probability that the noise voltage envelope will exceed the voltage threshold Vr is

00 R

( R2)

Probability

(Vr

< R <

oo)

=

J .,,

^{exp -}

₂ ^.,,

^dR

V1·'f'O 'f'O

(2.23)

(2.24) Whenever the voltage envelope exceeds the threshold, a target detection is considered to have occurred, by definition. Since the probability of a false alarm is the probability that noise will cross the threshold,

Eq. (2.24)

gives the probability of a false alarm, denoted

Pra.

The average time interval between crossings of the threshold by noise alone is defined as the false-alarm time

Tra,

where

Ji

is the time between crossings of the threshold Vr by the noise envelope, when the slope of the crossing is positive. The false-alarm probability may also be defined as the ratio of the duration of time the envelope is actually above the threshold to the total time it could have been above the threshold, or

(2.25)

OJ O'

E

0 ;-

~

v,

'6

C

~ t/'ci1

OJ .

n 0

<ii

>

C 1 <l

- - - T1r - - - c - - - 1 * + 1 _ _ _ ______,_.__

Threshold

Time--+-

Figure 2.4 Envelope of receiver output illustrating false alarms due to noise.

where tk and T,, are defined in Fig. 2.4. The average duration of a noise pulse is approximately the reciprocal or the bandwidth B, which in the case of the envelope detector is BIF. Equating Eqs. (2.24) and (2.25), we get

1 V}

Tra

= BIF exp 2t/lo (2.26)

A plot of Eq. (2.26) is shown in Fig. 2.5, with V} /2t/1₀ as the abscissa. If, for example, the bandwidth of the IF amplifier were 1 MHz and the average false-alarm time that could be tolerated were 15 min, the probability of a false alarm is 1.ll x 10-⁹_ From Eq. (2.24) the threshold voltage necessary to achieve this false-alarm time is 6.45 times the rms value of the noise voltage.

The false-alarm probabilities of practical radars are quite small. The reason for this is that the false-alarm probability is the probability that a noise pulse will cross the threshold during an interval of time approximately equal to the reciprocal of the bandwidth. For a 1-MHz bandwidth, there are of the order of 1Q⁶ noise pulses per second. Herice the false-alarm probability of any one pulse must be small ( < 10-⁶) if false-alarm times greater than 1 s are to be obtained.

The specification of a tolerable false-alarm time usually follows from the requirements desired by the customer and depends on the nature of the radar application. The exponential relatjonship between the false-alarm time

Tra

and the threshold level Vr results in the false- alarm time being sensitive to variations or instabilities in the threshold level. For example, if·

the bandwidth were 1 MHz, a value of 10 log (V}/2ij,_{0 )} = 12.95 dB results in an average false-alarm time of 6 min, while a value of 14.72 dB results in a false-alarm time of 10,000 h.

Thus a change in the threshold of only 1.77 dB changes the false-alarm time by five orders of magnitude. Such is the nature of gaussian noise. In practice, therefore, the threshold level would probably be adjusted slightly above that computed by Eq. (2.26), so that instabilities which lower the threshold slightly will not cause a flood of false alarms.

If the recdver were turned off (gated) for a fraction of time (as in a tracking radar with a servo-controlled range gate or a radar which turns off the receiver during the time of transmis- sion), the false-alarm probability will be increased by the fraction of time the receiver is not operative assuming that the average false-alarm time remains the same. However, this is usually not important since small changes in the probability of false alarm result in even smaller changes in the threshold level because of the exponential relationship of Eq. (2.26).

Thus far, a receiver with only a noise input has been discussed. Next, consider a sinc:-wave signal of amplitude A to be present along with noise at the input to the IF filter. The frequency

1 year

·· 6 months

1,000

30 days

_,:;;.

2 weeks h..~

(/) I week

E 100

'-0

0 3doys

<I)

U? 2 days

... 0 i::::

Q.) I day

Q.)

-

^Q.) 3

..0 _w 12 h

·.;::. E

(I) O"

0 '-

Q.)

""

<1

1 h

15 min 0.1..._~~ ... ~~ ... ~~..._~...,'---'----'~-'---'~~--'

8 9 10 . It 12 13 14 15

Threshold-to-noise ratio vrY211'o, dB

Figure 2.5 Average time between false alarms as a function of the threshold level V, and the receiver bandwidth B; t/1₀ is the mean square noise voltage.

of the signal is the same as the IF midband frequency

f.F.

The output of the envelope detec~or has a probability-density function given by⁹

R ( R²

+

^A2)

(RA)

ps(R) = t/Jo exp - 2t/Jo Io !po (2.27)

where /₀(2) is the modified Bessel function of zero order and argument Z. For Z large, an asymptotic expansion for I ₀(Z) is

I ⁰

(Z) ~ -& _v

_2n:Z

^{(1 +}

₈₂

^{l + · ··)}

When the signal is absent, A

=

0 and Eq. (2.27) reduces to Eq. (2.21 ). the probability-density function for noise alone. Equation (2.27) is sometimes called the Rice probability-density function.

The probability that the signal will be detected (which is the probability of detection) is the same as the probability that the enve]ope R will exceed the predetermined thresho]d VT. The

probability· of detection Pd is therefore Pd= ("'1,i(R)dR

' I' 1

·"" R ( R

²

+

A²)

(RA)

/ ---exp - 10 - · dR

' V r

t/t

^O

2t/t

O t/J 0

(2.28)