Fourier Transforms in Radar And Signal Processing_5 doc

Thus we obtain the Hilbert sampling theorem, which is more simply stated than the uniform sampling theorem for the same type of waveform: If a real waveform u has no spectral energy outs

ping spectrum This could be gated with a rectangular window of any width

from W to W′ to obtain V again Thus we obtain the Hilbert sampling theorem, which is more simply stated than the uniform sampling theorem

for the same type of waveform:

If a real waveform u has no spectral energy outside a frequency band

of width W centered on a carrier of frequency f0, then all the information

in the waveform is retained by sampling it and its Hilbert transform uˆ

at a rate W The samples are complex, the real parts are the samples of

u , and the imaginary parts are the samples of uˆ

We note that the sampling rate is independent of f0, unlike the casefor uniform sampling or quadrature sampling (an approximation to Hilbertsampling, described in Section 4.6 below) As pointed out by Woodward,

a real waveform of duration T and bandwidth W requires (as a minimum) 2WT real values to specify it completely—either real samples at a rate 2W

(as given by wideband sampling, or as the minimum rate in the case of

uniform or quadrature sampling) or WT complex samples (containing WT

real values in each of the real and imaginary parts) in the case of Hilbert

sampling The waveform can be said to require 2WT degrees of freedom

for its specification

4.6 Quadrature Sampling

4.6.1 Basic Analysis

If it is not convenient or practical to use a quadrature coupler or any othermethod to produce the Hilbert transform of a narrowband waveform, anapproximation to the transformed waveform can be obtained by delayingthe signal by a quarter cycle of its carrier frequency This follows from thefact that the Hilbert transform is equivalent to a delay of ␲/2 radians (forall frequency components, as shown in Appendix 4A), so the quarter cycledelay will be correct at the center frequency and nearly so for frequenciesclose to it The smaller the fractional bandwidth, the better this approximationbecomes As this is an approximation to the Hilbert transform, it follows

that sampling at the rate 2W (the Hilbert sampling rate) will not, in general,

sample the waveform adequately (to retain all the information contained init) However, we will see, below, that the method will in fact sample correctly,but at the cost, compared with Hilbert sampling, of requiring an increased

sampling rate, the rate depending on the ratio of bandwidth to centerfrequency (similar to the case of uniform sampling).

If u (t ) is the basic waveform, with spectrum U ( f ), then a delayed version u (t−␶) has spectrum U ( f ) exp (−2␲if␶) If we repeat the spectrum

of u at intervals W, corresponding to sampling at the rate W, we will obtain

an overlapping spectrum that, when gated, is not equal to U in general However, a suitable combination of the repeated spectra of u and its delayed version will give U after gating We start by imposing the condition 2f0 =

kW, where k is an integer, so that there is complete overlap of the two parts

of the spectrum of u , and also of the two parts of the spectrum of its delayed

version when repeated (Figure 4.10)

The appropriate identity for U is

U ( f )= 1

2{repW U ( f )+ exp (2␲if␶) repW [U ( f ) exp (−2␲if␶)]} (4.17)

⭈ [rect ( f− f0)/W+ rect ( f+ f0)/W ]

if ␶ is correctly chosen To check this identity, we consider the output of

the positive frequency spectral gate for frequencies in the range f0 − W /2

< f < f0 + W /2 In this interval we have, as there is overlap of the tive frequency part of the spectrum, moved up by 2f0, or kW, for some integer k ,

This is simply U ( f ), as required, if we choose␶ such that 4f0␶=1,

or, more generally, if 4f0␶ = 2m + 1, where m is an integer The same

condition results if we consider the output of the negative frequency gate—

we simply replace f0with−f0 throughout Thus, the required delay is seen

to be an odd number of quarter wavelengths of the carrier, or center frequency

f0, or one quarter cycle in the simplest case Taking the (inverse) Fourier

transform of the identity for U ( f ) in (4.17), we have

u (t ) =1

2 [(1/W ) comb 1/W u (t ) +␦(t +␶)

⊗ (1/W ) comb 1/W u (t −␶)]⊗ 2W␾(t ) (4.19)

=comb1/W u (t )⊗␾(t ) +comb1/W u (t −␶) ⊗␾(t+ ␶)where ␾ is the interpolating function This is obtained from the (inverse)Fourier transform of the spectral gating function ⌽, defined by

2W ⌽( f ) = rect [( f− f0)/W ]+ rect [( f +f0)/W ] (4.20)Thus,

2W␾(t )= W sinc (Wt ) [exp (2␲if0t )+ exp (−2␲if0t )]

or

␾(t )= sinc (Wt ) cos (2␲f0t ) (4.21)This interpolating function also appears in the uniform sampling case[see (4.9)] and the Hilbert sampling case [see (4.16)] Equation (4.19) states

that the real waveform u is equal to the sum of the waveform obtained by sampling u at intervals 1/W (i.e., at rate W ) and interpolating with the

function␾and the waveform obtained by sampling a quarter-wave delayed

version of u and interpolating with a quarter-wave advanced version of␾

To remove the condition relating W and f0, we choose W′ ≥W such that 2f0=kW′, where k=[2f0/W ], the largest integer in 2f0/W We then repeat the spectrum at intervals W′, which corresponds to sampling at the

rate W′, but we can keep the same spectral gating function and hence thesame interpolating function The minimum required sampling rate, relative

to the minimum rate, equal to the bandwidth W, is r =W′/W =1 +␣/k

if 2f0/W =k +␣ This minimum rate is plotted in Figure 4.11, and this

is the rate given by Brown [2]

If W′is increased to higher values such that 2f0=nW′for n integral,

n<k , we again obtain sampling rates that will retain the waveform

informa-tion, and these are shown by the dashed lines in Figure 4.11 The required

sampling frequency could be obtained in practice by synchronizing W′ to

a submultiple of 2f0(ideally the k th, for the minimum rate).

4.6.2 General Sampling Rate

Unlike the uniform sampling case, the required sampling rates determined

so far are precise (Figure 4.11) instead of within bands (as in Figure 4.8)

This is because the delay has been chosen to be a quarter cycle of f0 (or an

odd number of quarter cycles) In fact, on replacing 2f0with kW′in (4.18),

where kW′is the frequency shift that takes U−, centered at −f0, onto U+,centered at+f0, we see that the condition to be satisfied is 2kW′␶=2m+1

(m an integer) If we relate the delay␶ to the sampling rate W′ instead of

directly to f0, then we have more freedom of choice of W′ In Figure 4.12(a),

we see part of the function repW U−, the signal band at −f0 repeated at

intervals W, in the region of+f0where 2f0is not an integer multiple of W.

If we consider the part of this spectrum that overlaps the band of width W,

centered at+f0, we see that there is a mixture of parts of U−shifted by kW and by (k+1)W If the delay is correct to make U−disappear when shifted

by kW, then it is not quite correct when shifted by (k+ 1)W, and a small

amount of spectral overlap occurs

Figure 4.11 Relative sampling rates (basic quadrature sampling).

Figure 4.12 Shifted positions of U−: (a) 2f0=(k+ ␣)W, (0< ␣ <1); (b) 2f u=(k+1)W′ ;

and (c) 2f1=(k−1)W′

The minimum repetition rate to avoid this is shown in Figure 4.12(b),

where W′ (> W ) is such that (k+ 1)W′ moves U− just beyond the gated

region (between f1 and f u ) Because W′ > W gaps of width W′ − W now occur between the repeated versions of U− The minimum required value

of W′is given by (k +1)W′ =2f u (In fact, other local minimum rates are

given by W′ such that (n+1)W′ =2f u , for n integral n< k , but we will

see that we do not need to consider these rates because of a more general

result below.) In this case, in order for U−to disappear in the gated band,

the delay must satisfy 2kW′␶=1, and so, with the condition on W′above,

we find that␶=(k+1)/4kf u—that is, the delay should be (1+1/k ) times

a quarter cycle of the upper edge of the signal band f u(or an odd multiple

of this)

If we increase the sampling rate further, we reach the condition shown

in Figure 4.12(c), where the band U− has just reached the lower edge of

the gated band This is when (k − 1)W′ = 2f l (or, again, more generally

when (n −1)W′ = 2f l for n an integer and n ≤ k ) The delay required is

(1 − 1/k ) times a quarter cycle of the lower edge of the signal band f l (or

an odd multiple of this)

To summarize, the minimum and maximum relative sampling rates

are given by r m=2f u /W (n+1) and r M=2f l /W (n− 1), where f u =f0 +

W /2 and f l=f0−W /2; a central rate (very close to the mean of these two)

is r c =2f0/nW With k and ␣defined by 2f0 =k +␣, these become r m =

(k+␣ +1)/(n +1), r M= (k+␣− 1)/(n− 1), and r c= (k+␣)/n , where

n ≤ k (n and k integers and 0≤ ␣ <1) When n =k, these become r m =

1 + ␣/(k + 1), r M =1 + ␣/(k− 1), and r c =1 + ␣/k Also, when␣ = 0

(2f0/W integral), then these rates are all unity, and n < k corresponds to the continuations of these particular lines from lower k values, as illustrated

in Figure 4.13

The allowed sampling rates relative to the bandwidth W are given in the shaded areas in Figure 4.14 The maximum and minimum rates r Mand

r m define the boundaries, and the central values rc are shown as dashed lines

in Figure 4.14 We note from Figure 4.14 that there are no unallowed

sampling rates above 2W This is because when the interval between tions of U− becomes 2W, it is not possible to have parts of more than one repetition of U− in the gating interval (see Figure 4.12(b or c) with W′ ≥

repeti-2W ), so if the delay is correctly chosen, the U−contribution in this interval

can always be removed [By putting x=f0/W=(k+␣)/2 and equating r m

at n =k −1 and r m at n = k , with␣ = 2x −k , we find these lines meet

at x =k − 1⁄2 and the common value of r is 2, as shown in Figure 4.14.]

However, the general rates given in Figure 4.14 may not be very

convenient in practice, as they require choosing the delay to be 1/2kW′

[where W′is between 2f u /(k + 1) and 2f1/(k −1)], which may not be as

easy as choosing it to be 1/4f0, as assumed in Figure 4.11 Because therequired delay is no longer exactly a quarter cycle (or an odd number of

quarter cycles) of the carrier, this sampling has been termed modified ture sampling in the title of Figure 4.14 In fact, the central rate 2f0/k does

quadra-require the quarter cycle delay, but from this study we see that this is notthe minimum rate, if that is what is required

Figure 4.13 Lines of relative sampling rates.

Figure 4.14 Relative sampling rates (modified quadrature sampling).

Thus, we can now state a quadrature sampling theorem:

If a real waveform u has no spectral energy outside a frequency band

of width W centered on a carrier of frequency f0, then all the information

in the waveform is retained by sampling it and a delayed version of it

at a rate given by rW, where r is given in (4.22) and the delay (which

is close to a quarter cycle of f0) is given in (4.23) The samples are

complex, the real parts are the samples of u , and the imaginary parts

are the samples of the delayed form.

4.7 Low IF Analytic Signal Sampling

A signal u (t ) on a carrier at frequency f0can be written u (t )=a (t ) cos [2␲f0t

+␾(t )], and, at least in principle, we can derive its Hilbert transform, uˆ (t )

= a (t ) sin [2␲f0t + ␾(t )], and hence the complex form u (t ) + iuˆ (t ) =

a (t ) exp i [2␲f0t+␾(t )] The information in this signal is contained in the amplitude and phase functions a (t ) and␾(t ), and what is required for digital signal processing is a digital form of the analytic signal a (t ) exp i␾(t ) This

is what is given by Hilbert sampling and quadrature sampling, discussed

above, in particular from the point of view of finding the minimum samplingrate needed to preserve all the signal information An alternative method ofobtaining the sampled analytic, or complex baseband, signal is given in thissection This is simpler to implement in practice—not requiring the Hilberttransform or an accurate quarter cycle delay, and sampling in only a singlechannel rather than in two—at the cost of requiring a higher sampling rate,though, at the minimum, this single sampling device [or analogue-to-digitalconverter (ADC)] operates at just twice the rate of the two ADCs neededfor the alternative methods.

The method requires bringing the signal carrier frequency down fromthe normally relatively high RF to a low IF To avoid the two parts of the

spectrum overlapping, we see that we must have f0≥W /2 The samples we require are those corresponding to the complex baseband waveform V ( f ),

F comb1/F [2u (t ) exp (−2␲if0t )]⊗W sinc Wt (4.24)

Figure 4.15 Low IF sampling spectra.

v (t ) =2W

F combT [u (t ) exp (−2␲if0t )] ⊗sinc Wt (T= 1/F )

Thus the analytic, complex baseband waveform is given by sampling the

real IF waveform u multiplied by the complex exponential exp (−2␲if0t )—

that is, mixing down to baseband using a complex local oscillator (LO) at

the signal’s center frequency f0 (Again, in principle, to form this waveform,

we interpolate the samples obtained at intervals T=1/F, where the sampling rate F is 2f0 +W or higher, with sinc functions.) In fact, we do not have

to provide this LO waveform in continuous form, as we note that

we just multiply the real samples of u by 1,−i ,−1, and i in turn, a particularlysimple form of down-conversion

If the IF is greater than W /2 (up to 3W /2), then we can repeat the spectrum at the smaller interval of 2f0 + W, rather than 4f0, but in thiscase the complex down-conversion factors are not so simple (being given byexp [−␲in /(1 + W /2f0)]) and generally the 4f0 sampling rate will be pre-ferred If the IF is considerably higher than the bandwidth, then lowersampling rates that avoid overlapping can be used, as discussed in Section4.4, under the topic of uniform sampling, of which this method is anexample Using the notation of Section 4.4, the lowest IF case corresponds

to f u =W and k= 1 For higher IF values we have f u= f0+ W /2=kW′,

where W′is the lowest value above (or equal to) W such that f u /W′ is an

integer k Then the minimum required sampling rate is 2W′ =(2f0+W )/k ,

and the complex down-conversion factors are exp (−2␲if0nT ), where T =

1/2W′, leading to the factors exp [−␲ikn /(1 + W /2f0)] Again, this is anawkward form to apply, but if we chose the slightly higher sampling rate

of 2f0/(k−1⁄2), as suggested in Section 4.4, then the down-conversion factorsbecome simply exp [−␲in (k − 1⁄2)] or −i n for k odd and i n for k even.

However, sampling with a finite window width on a high IF may require

care, as discussed in the next section, and keeping the IF low would generally

be preferable

4.8 High IF Sampling

If we sample at a relatively high IF, the time taken to obtain a sample ofthe waveform may become significant compared with the period of thecarrier We take for our model a device that integrates the waveform over

a short interval␶, the sample value recorded being the mean waveform valueover this interval, the integral divided by ␶ We see that this value is thesame as would be given by a device that sampled instantaneously the waveformgiven by sliding a (1/␶) rect (t /␶) function across the waveform and integra-ting—that is, forming the convolution of the waveform with the rect function

Thus, if u is the waveform, the samples actually correspond to the waveform

v given by

v (t )= u (t )⊗(1/␶) rect (t /␶) (4.26)The spectrum of this is

V ( f )=U ( f ) ⭈ sinc ( f␶) (4.27)

Figure 4.16 shows the spectrum of V compared with that of U, shown

as a rectangular band (in the positive frequency region only) With a low

Figure 4.16 Spectrum of waveform sampled with a finite window.

carrier frequency f0, compared with 1/␶(i.e., with␶a small fraction of theperiod of the carrier), in position ‘‘a,’’ there is relatively modest distortionacross the signal band At a higher center frequency, position ‘‘b,’’ and alsowith a larger bandwidth, the distortion is more serious At position ‘‘c,’’

where the window is one cycle of f0( f0␶ =1), the distortion is severe andtotally unacceptable However, in position ‘‘d,’’ where the sinc function is

near a stationary value, the distortion is very low This is at f0␶ = 1.434,

so the window␶should be about 1.4 cycles of the carrier for a low distortionresult

4.9 Summary

In this chapter we have shown how the rules-and-pairs method can be used

to obtain some sampling results very neatly and concisely The main aimwas to determine the minimum sampling rates that would retain the signalinformation, but in some cases the method was used to find what otherrates would be acceptable (not necessarily all rates above the minimum).This was first applied to sampling wideband signals, with significant spectral

power from some maximum W down to zero frequency The information

in a real waveform is all retained by sampling it at the rate 2W (or any

higher rate) The second example, uniform sampling, applies to a narrowbandsignal, a signal on a carrier, with a spectrum limited to a band of frequenciesaround the carrier In this case, the rates acceptable are dependent on the

ratio of the bandwidth W to the center frequency f0 being at least 2W and

generally higher This form of sampling is an example of the case wheresome higher sampling rates are not allowed if distortion is to be avoided

A different approach is to convert the real waveform into the complexwaveform that has the given waveform as its real part This requires derivingthe imaginary part from the real part by means of a Hilbert transform Inprinciple, this is applicable to both wideband and narrowband waveforms,though it is more likely to be applied to the latter in practice Given thecomplex waveform, we find very quickly that we only have to sample (in

the two channels, real and imaginary) at the rate W (or any higher rate) to

obtain complex samples representing the waveform It is this complex formthat is normally required for digital signal processing

Hilbert sampling seems a very satisfactory approach, but it does depend

on the provision of a good Hilbert transform, which is equivalent to awideband (all-frequency) phase shift of 90 degrees A close approximation

to Hilbert sampling for narrowband waveforms is quadrature sampling, where