The effect of filtering on the Signal-to-Noise Ratio (SNR) of a coherently demodulated band-limited signal is determined in the presence of worse-case amplitude ripple. The problem is formulated mathematically as an optimization problem in the L2-Hilbert space. The form of the worst-cast amplitude ripple is specified, and the degradation in the SNR is derived in a closed form expression. It is shown that when the maximum passband amplitude ripple is 2 delta (peak to peak), the SNR is degraded by at most (1 - delta squared), even when the ripple is unknown or uncompensated. For example, an SNR loss of less than 0.01 dB due to amplitude ripple can be assured by keeping the amplitude ripple to under 0.42 dB

Hurd, W.

Sadr, R.

English

NASA Technical Reports Server

TDA Progress Report 42-88
N87-17952
October-December 1986
Filter Distortion Effects on Telemetry Signal-to-Noise Ratio
R. Sadr and W. Hurd
Communications Systems Research Section
The effect or filtering on the Signal-to-Noise Ratio (SNR ) of a coherently demodulated
band-limited signal is determined in the presence of worst-case amplitude ripple. The
problem is formulated mathematically as an optimization problem in the L2-Hilbert
space. The form of the worst-case amplitude ripple is specified, and the degradation in the
SNR is derived in a closed form expression. It is shown that when the maximum passband
amplitude ripple is 2A (peak-to-peak), the SNR is degraded by at most (1 - A2), even
when the ripple is unknown or uncompensated. For example, an SNR loss of less than
O. O1 dB due to amplitude ripple can be assured by keeping the amplitude ripple to under
0.42 dB.
I. Introduction
Amplitude ripple is inherent in most physically realizable
filters. We seek to determine the system performance degrada-
tion resulting from amplitude ripple. The Signal-To-Noise
Ratio (SNR) is used as the performance criterion. Our results
provide an easy method to determine the worst-case loss due
to amplitude ripple. To derive the expression for the worst-
case loss in the SNR, the class of worst-case amplitude ripple
is explicitly found.
We consider filtering a signal y(t) which is composed of a
bandlimited signal s(t), with bandwidth (fo- W, fo + If)
Hertz, added to a noise process (n(t)}, where fo denotes the
center frequency and W is the half bandwidth of the signal.
The passband of the filter covers the same band of frequency
as the signal s(t). This filter is shown in Fig. 1. Ideally, the
transfer function for the filter is
Ho(/_ ) = d_"
for I_ot e (2rr(f o - W), 27r(f o + W)) where r is a constant group
delay. The shape of the filter response at frequencies outside
of the signal band is shown to be immaterial.
The amplitude spectrum of the ideal filter, H o qeo), has con-
stant gain and group delay in the passband frequency. This is
never really true in practice where the gain of the filter ex-
hibits a bounded ripple in the passband. The amplitude ripple
is the deviation of IHo(/_o)t from ideal and is denoted by
A(jw). Characteristics of both an ideal and a nonideal band-
pass filter are depicted in Fig. 1. The transfer function of the
nonideal filter in the passband is represented as
Ho(Jco ) = (1 + a(./w)) e/(¢(_)+_T) , (1)
for Icole (2rr(f o - W) , 27r(f o + W)) where ¢(jw) represents
the deviation in phase from constant group delay, r.
A critical issue in the specification of the filter h(t), for the
design engineer, is to determine the impact of A(/'w), and
59
https://ntrs.nasa.gov/search.jsp?R=19870008519 2020-03-20T12:40:11+00:00Z
¢(jco) on the SNR. The degradation of SNR for phase devia-
tion ¢(jco) has been studied previously. J. Jones (Ref. 1) in
1972 analyzed the filter distortion effects of the phase non-
linearity for BPSK and QPSK, and has shown that when phase
deviation ¢(jco) is bounded by emax in absolute value, the SNR
is degraded by at most a factor of cos2q_max.
In this article, it is shown that if the amplitude ripple is
bounded by A (i.e., [A(ja))l_< A), the SNR in the presence of
the worst case amplitude ripple waveform is degraded by at
most (1 - A2). This result holds even when the amplitude
ripple is unknown, or known but not compensated.
To summarize the outline of the rest of this article, Sec-
tion II describes the system under study and the underlying
assumptions for which the SNR figure is analyzed. In Sec-
tion III, the closed form SNR expression is derived for a
coherently demodulated signal which is filtered by the non-
ideal transfer function characteristic defined in Eq. (1). In
Section IV, the class and properties of the worst-case ampli-
tude ripple are specified. Finally, in Section V, we make some
concluding remarks based on our results.
II. Formulation
We consider a received waveform containing signal and
noise, that is, y(t) = s(t) + n(t). The signal amplitude spectrum
S(jw) is band-limited to If1 e [fo + W, fo - W], and its wave-
form is completely known during each te[O,T]. The noise
process {n(t)} in our analysis is assumed to be an Additive
White Gaussian Noise (AWGN) process with single-sided spec-
tral density N o W/Hz. The results generalize to the case where
the noise is not white. The only restriction is that {n(t)}be a
wide-sense stationary process. This implies that it has zero
mean and autocorrelation function R n (r) = E [n (t) n (t + z)],
where E [ .] denotes the expectation operator.
The optimal receiver for the observed signal y(t), which
maximizes the SNR, is a matched filter (Ref. 2). This solution
is expressed in the form of the Fredholm integral equation of
the first kind. There are known methods to solve this integral
equation explicitly to find the optimal matched filter solution
hMX(t).
For an AWGN channel the matched filter solution is
hMA (t) = s(T - t) or, equivalently, in the frequency domain it
is HMA (JCO) = S*(jco) e-]wT. (Throughout the article, super-
script * denotes complex conjugate while a midline * denotes
convolution).
For the case in which the noise is only wide sense station-
ary (not necessarily AWGN) with spectrum Sn(Jco), the
matched filter solution may be expressed under certain
assumptions (Ref. 2) as HMA (/cO)= S*(/CO)e-ic°T/Sn(JCo).
This transfer function is recognized as the matched filter
transfer function for the white noise case divided by the actual
power density of the noise. Therefore, it is possible to gen-
eralize our result for wide sense stationary noise processes by
simply using a matched filter which is matched to both the
noise and the known signal. Thus, with no loss of generality,
in our subsequent analysis we assume that the noise is white
and Gaussian.
In digital communication systems, the signal s(t) is modu-
lated at the transmitter to a Radio Frequency (RF) by multi-
plying s(t) by the carrier signal cos(co0t), where 2n coo is the
carrier trequency. At the recewer (Fig. 2), the observed signal
is filtered and then demodulated by multiplying the observed
signal by 2 cos(co0t ). This signal is passed through a zonal
low-pass filter to filter out the double frequency terms pro-
duced by the multiplication operation. The output of the
low-pass filter is then fed to the matched filter.
The sampled output of the matched filter each T s is de-
noted by M i.
The SNR is defined as the ratio of the square of the ex-
pected value to the variance of the random variable M i. In the
following section, a closed form expression for the SNR is
derived. This expression is formulated in the form of a func-
tional. We minimize this functional over the ensemble of all
possible amplitude waveform ripples in the passband of the
filter hBF(t), as shown in Fig. 1.
III. Signal-to-Noise Ratio Expression
Since the filtering processes are linear, we can consider the
system response to signal and noise separately. We need to
determine the mean value of the signal and the variance of the
noise, both at the matched filter output. We denote the
response of each stage of the system to the signal by ei(t),
as shown in Fig. 2. The signal is represented by amplitude
spectrum throughout the following analysis. Throughout this
article, the square of a complex function is meant to be the
magnitude square of that function.
Neglecting the noise response, the amplitude spectrum of
y(t), Y(]co), is expressed as
r(jco) = s(j(co - %)) + s(j (co+ coo)) (2)
We denote the bandpass filter H'(jco)as
Y'(/co) = HBF(](_o- COO))÷H_AJ('-_ +%))
6O
where HBF (o) is the transfer function of a complex low-pass
filter. The spectrum of e l(t), the output of the bandpass
filter under study (Fig. 2), is
El(/Co ) = H'(jCo) Y(jco) (3)
The demodulated waveform e2(t ) has the spectrum containing
the sum and difference frequencies,
E2(/Co ) = El(/(Co - Coo) ) +El(/(Co + 030) )
= 1¢(j(co - %)) Y(/(co - %))
+ H'(j(CO + coo)) Y(j(co+coo) ) (4)
The demodulator is followed by an ideal low-pass filter, which
filters out the double frequency terms, and the resulting out-
put spectrum Ea(/co ) is E3 (/co) = E2(Jco ) HLp(jco), where
t 1, Ico I_< 21r(f o - W)
HLp(JCo)
0, otherwise
Thus, the output of the low-pass filter from Eq. (4) is
E3(jco ) = (H_F(-jco) + HBF(jco)) S(jco ) (5)
Note that if the low-pass filter is not ideal, its deviation from
ideal should be included in the filter under study.
Neglecting deviation in HMA(jco ) from ideal, the matched
filter that maximizes the SNR has the transfer function
(Ref. 2) hMA(t ) = s(T- t), or equivalently
HMA (jCo) = S* (]co) e-]cot (6)
Let for simplicity H (jco) denote H_y(-jco ) + HBF(]CO). Then
the matched filter output e4(t ) can be represented as
F4(/Co) = E3(/Co) s* (/Co) e-jcor
and substituting Ea(jco ) using Eq. (5), e4(t ) may be expressed
as
e4(t ) = _ (jco) IS(jco)l 2 e-jt°(T-t) dco (7)
where I = I-IV, W]. The output of the matched filter is sampled
at the end of every time interval T. Thus at t = Twe have
1 rlit(/co). IS(jco)l 2 dw
e4(:r ) = J,
The system noise response is denoted by z(t). The random
process {Mi} (taking values in IR1) is the sum of the filtered
signal plus the filtered noise component. Hence, we can write
M. = e4(r)+z(r) (8)
Taking expectation of Eq. (8), and noting that the noise is
assumed to be zero mean, we get
E[Mi] = _ (jw) IS(/¢_)I = d_ (9)
To compute Var[Mi], let z(t) = n(t) * x(t). Note that z(t) is
a filtered white noise process, which is filtered by the filter
under study and the matched filter. The cascaded filter is
denoted by x(t).
Let X (jco) = S (jCo) H ( jo_) e -/c°T, thus we have
Var [M_.I = Var [z (t)] ] t=r (lO)
and
z (t) = n(t) • x(t)
E[n(t) n(t + r)] = Nz 6(r)
(11)
From Eq. (11), the variance ofz(t) can be expressed as
No j_X(/Co)12Var[z(t)l = _- deo
I
(12)
Combining Eqs. (12) and (10) and evaluating these expressions
at t = T, results in
Var [Mi I No= -_ IH(j_)S(jCo)l 2 dco (13)
Therefore since SNR = (E[MI. ] )2 / Var [Mi] we have
2 (fzH(Jco)S2(Jc°)&°) 2
= -- (14)
SNR No /_lH(/co) S(/Co)12 do_
61
@ .
The filter transfer function is H(fl.o) = HBF(JCo ) + H_F(-Ico )
and furthermore H(jco) is
H(j¢o) = (1 + A(j_)) eJ_r ¥ l¢ol _<2_'l¢
We assume that the group delay is negligible (i.e., ejwr _ 1), or
it is compensated in the matched filter, hence, by substituting
H(jw) in the SNR expression (14), the SNR is
2 ]fs(l+2x(Jc°))S2(jw)dwl2
SNR - No fl (1 + A(JCO)) 2 $2(J6°) d6° (15)
IV. Worst Case Amplitude Ripple
With no loss of generality assume I = [0,1 ] and let
o
Using Eq. (15)we can formulate the minimization problem
F(g) = inf
g(DeL2[o,II
(i I (1 +g(_))d_)
(16)
subject to the constraint
Ig(_)[ < A < 1 (16a)
The integrals are understood in the Lebesgue sense.
In the appendix we prove the following theorem. In the
proof, the class of the functions for which the minimum
occurs is explicitly exhibited.
Theorem 1. There is a continuum of measurable step func-
tions which minimize Eq. (16), and at the minimum FMIN@ ) =
1 -- 4 2, and g(_) is
/A' for0<_<12-A-
_(_) } 1-A
_,-A, for _ <_< 1
Using the result of Theorem 1, we can state the following
corollary.
Corollary 1. Define interval I_, as any subset of the inter-
val I such that
fzl S2(jw) dw - 1 - A2
Then Eq. (15) is minimized by
/x (/co)= {_'
for 6oeI 1
I,-A, for co ¢ 11
Corollary 2. The minimum SNR for Eq. (15) is achieved by
A(j6o) of the form specified by corollary (1), and further-
more, the total SNR for the worst case ripple distortion is
(1 - A 2) SNRiaea 1.
Corollaries (1) and (2) are direct consequences of Theorem
1. The point a, in Fig. 3, indicates the point at which the inte-
grated power of the signal s(t) is (1 - A)/2.
In general, the SNR is minimized when the ripple is +A for
frequencies containing (1 - A)/2 of the energy of s(t), and
_A for frequencies containing (1 + A)/2 of the energy of
s(t).
The shape of the amplitude ripple is not unique, and it is
the whole continuum of step functions which satisfies the con-
dition stated in corollary (I). To construct another amplitude
ripple waveform which satisfies the conditions of corollary
(1), one can take the waveform of Fig. 3 and move a segment
from [0,a] to (a,1], and move an equal energy segment from
(a,l] to [0,a]. Conceptually, this method may be thought of
as juggling equal energy line segments from each interval. This
method results in obtaining a new step function from the
basic function of Fig. 3 which satisfies the statement of
corollary (1).
V. Discussion and Conclusion
We have shown that the worst-case loss in SNR due to
amplitude ripple is (1 - A2). This result can be used to specify
filters for communication receivers, such as the Advanced
Receiver for the NASA's Deep Space Network.
To express the maximum allowable ripple, for a given loss
(LaB) in dB, we expand the expression for dB of ripple and
62
keep the first terms. Thus for small ripple and loss, the ripple
loss in dB may be expressed as
RippledB (LdB) = 4.168 LX/_dB
Consider a simple example: If the system can tolerate a loss
of 0.01 dB due to amplitude ripple, the maximum allowable
ripple in dB would be 0.4168 dB. The loss is much less than
the ripple and decreases as the square of the ripple.
From a mathematical standpoint, we solved the minimiza-
tion problem stated in Eq. (16). And we showed that the opti-
mal solution lies on the boundary, and it is a continuum of
measurable step functions in the interval [0,1]. This is expli-
citly exhibited in the appendix.
Acknowledgment
Special thanks are due to Dr. Mehrdad Shahshahani for his help in proving theorem 1
and Dr. Eugene R. Rodemich for initially pointing out the solution.
References
1. Jones, J. J., "Filtered Distortion and Intersymbol Interference Effects on PSK Signals,"
IEEE Trans. Comm. Tech., COM-19,No. 2, April 1971: 121-132.
2. Van Trees, H. L., Detection, Estimation, and Modulation Theory. New York: John
Wiley & Sons, Part I, 1968;Part II, 1971.
63
S(t) + n(t) y(t)
2Tr(f 0 - W) 2_rf 0 2Tr(f(
_11111
+ W)
Fig. 1. Filtering
y(t)=S(t)+n(t)r hBF(t)BANDPASSFILTERf0
2 cosu) 0t
LOW PASS
FILTER
hLp(t)
e3(t)
Fig. 2. Communication model
MATCHED
FILTER
hMa(t) = S(T - t)
e4(t)
I+A
I
I-A
a
UIIIUIIU
Fig. 3. Ripple waveform
r (_/27rW
64
Appendix
Let0<A< 1 and
Ca = { f: [0,1] + IR If measurable
and If(z) - I I< A for almost all u
We equip Ca with the induced metric from LI(I) (1 = [0,1] ) so
that Ca becomes a complete metric space (we identify func-
tions equal almost everywhere). Consider the continuous func-
tional F on Ca defined by
F(f) =
(fo1
1
foo f2(x) ax
In the appendix we investigate the existence and nature of
minima ofF. Let
C
n
{ (ao ..... a; c1 ..... c) I
0 = ao<_a l<_...<_a =l,cie [1-A,I+A]}
Then Cn is naturally identified with a compact subset of
[0,1] n-1 X [1 - A, 1 + A] n. To every 7 e Cn we assign the
function fir e C.r which takes value ci on the open interval
(ai_ 1 , ai) provided it is nonempty. (Note that f'r is undefined
on a finite set, and can be assigned arbitrary values there.)
The mapping
is a continuous function on Cn and hence achieves a minimum
on C n .
Lemma. Let 3" = (ao ..... an; c, ..... cn) and assume F(fy)
is a minimum on Cn. Then ci = 1 -+A whenever ai_ 1 < ai"
Proof. Let C'n+2 be the compact subset of C+2 consisting
of (ao ..... an+2" c 1 ..... cn+2) such that
c1 = l+A and cn+ 2 = 1-A
t
We embed Cn in Cn+ 2 by
(ao ..... an ; c1 ..... cn)
_(ao,a o ..... an_l,an,an; 1 + A, c,.... , c, 1 -A)
and prove the assertion for C'n+2.Let
1
£ (x)dx
M =
:t ( )dx
Assume Jet assumes value c 4= 1 -+ A on a nonempty open inter-
val (ai_l, ai). Either c _>M. r or c <M r and assume for definite-
ness that the former possibility occurs. We may assume c2 =
C _>M-t. In fact it is easy to see that there is 3" e Cn+2 such that
M,,/= My, F(f,_)= F(fr,) and c'2 = c where 3" = (ao,a',.....
a'n+2; C;,..., Cn+2). Let 0_<8 _< la2 -all and define 0 e C'n+ 2
by
O = (ao, a1 + 8,a 2 ..... an+2; 1 + A, C2 ..... Cn+l, 1 - A)
Then by a simple computation
A+ix
F(:®) - B + 3
where
/=
and
IX
m
1 (1 +A- c)6dx +2
¢ ++(1 + A- c)
Since c 1> My, by taking 6 > 0 sufficiently small we can ensure
a/3 < A/B and consequently F(f®) < F(ft). This proves the
lemma.
65
Let 0On}be a sequence in Ca such that
lira F(fn) = inf F(f)
nt** fec a
Since
is dense in CA there is sequence (3'k}with 3'ke Cnk such that
1
F(/_k) < F(fk) +?
In view of the lemma we may assume fvk takes only values
1 +- 2x and hence there is 0k e C2 such that F(f._k ) = F(f%).
Therefore inf F(f) is achieved for some F(fo) with ® =
(0, a, !; 1 + A, 1 - A). It is a simple exercise to show that a =
(1 - eX)/2 and F(fo) _- 1 - A2.
Finally we note that for any partition I = E 1 kJE 2 with
meas(E1) = (1 - A)/2, meas(E2) = (1 + A)/2 the function
= II+A ifxeE 1f(x)
! l-A, ifxeE 2
is also a minimum for F. The argument above also shows that
all minima ofF are of this form.
86


Filter distortion effects on telemetry signal-to-noise ratio

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