Reply by robert bristow-johnson January 1, 20072007-01-01
Oli Charlesworth wrote:
> On Dec 29, 11:20 am, "Rune Allnor" <all...@tele.ntnu.no> wrote: > > > > > > > X(z) = 1/(1-az^-1); provided |az^-1|<1 => |z| >|a| > > > > > ......(eq 2) > > > > > > > now X(z) can be written as X(z) = z/(z-a). > > > > > ......(eq 3) > > > > > equation 3 of X(z) is obtained by multiplying and dividing numerator > > > and denominator of X(z) of equation 2 with z. > > > > > It is the basics of algebra that you cant do it when z = 0. > > > > > 1/(1-az^-1) is not equal to z/(z-a) at z = 0. > > > > > so evaluation of the poles and zeros should be done for equations > > > expressed in negative powers of z and not positive powers of z > > > > > X(z) = 1/(1-az^-1); doesnt have zero at z = 0, Infact it doesnt have > > > any zeros > > > > > In conclusion "no zeros exist outside the Region of convergence"What you are say, then, is that > > > > X(z) = 1/(1-az^-1) = 1/(1-az^-1) *1 = 1/(1-az^-1) *z/z = z/(z-a) =/= > > X(z)... > > At z=0, indeed. In the same way that: > > 1 = 1 * z/z = z/z =/= 1 at z = 0
but when this is a removable singularity, i think it *is* accurate and appropriate to say something like " X(z) = 1/(1 - p*z^(-1)) has a zero at z = 0. " or simply that X(z) = 1/(1 - p*z^(-1)) = z/(z-p) = (z-q)/(z-p) where q = 0 they are, in fact, two different expressions of transfer function for the same LTI system. r b-j
Reply by Oli Charlesworth December 31, 20062006-12-31
On Dec 29, 11:20 am, "Rune Allnor" <all...@tele.ntnu.no> wrote:
> kee skrev: > > > > > > > X(z) = 1/(1-az^-1); provided |az^-1|<1 => |z| >|a| > > > > ......(eq 2) > > > > > now X(z) can be written as X(z) = z/(z-a). > > > > ......(eq 3) > > > equation 3 of X(z) is obtained by multiplying and dividing numerator > > and denominator of X(z) of equation 2 with z. > > > It is the basics of algebra that you cant do it when z = 0. > > > 1/(1-az^-1) is not equal to z/(z-a) at z = 0. > > > so evaluation of the poles and zeros should be done for equations > > expressed in negative powers of z and not positive powers of z > > > X(z) = 1/(1-az^-1); doesnt have zero at z = 0, Infact it doesnt have > > any zeros > > > In conclusion "no zeros exist outside the Region of convergence"What you are say, then, is that > > X(z) = 1/(1-az^-1) = 1/(1-az^-1) *1 = 1/(1-az^-1) *z/z = z/(z-a) =/= > X(z)...
At z=0, indeed. In the same way that: 1 = 1 * z/z = z/z =/= 1 at z = 0 -- Oli
Reply by Rune Allnor December 29, 20062006-12-29
kee skrev:
> > > X(z) = 1/(1-az^-1); provided |az^-1|<1 => |z| >|a| > > > ......(eq 2) > > > > > > now X(z) can be written as X(z) = z/(z-a). > > > ......(eq 3) > > > > equation 3 of X(z) is obtained by multiplying and dividing numerator > and denominator of X(z) of equation 2 with z. > > It is the basics of algebra that you cant do it when z = 0. > > 1/(1-az^-1) is not equal to z/(z-a) at z = 0. > > so evaluation of the poles and zeros should be done for equations > expressed in negative powers of z and not positive powers of z > > > X(z) = 1/(1-az^-1); doesnt have zero at z = 0, Infact it doesnt have > any zeros > > In conclusion "no zeros exist outside the Region of convergence"
What you are say, then, is that X(z) = 1/(1-az^-1) = 1/(1-az^-1) *1 = 1/(1-az^-1) *z/z = z/(z-a) =/= X(z)... Rune
Reply by kee December 28, 20062006-12-28
> > X(z) = 1/(1-az^-1); provided |az^-1|<1 => |z| >|a| > > ......(eq 2) > > > > now X(z) can be written as X(z) = z/(z-a). > > ......(eq 3) > >
equation 3 of X(z) is obtained by multiplying and dividing numerator and denominator of X(z) of equation 2 with z. It is the basics of algebra that you cant do it when z = 0. 1/(1-az^-1) is not equal to z/(z-a) at z = 0. so evaluation of the poles and zeros should be done for equations expressed in negative powers of z and not positive powers of z X(z) = 1/(1-az^-1); doesnt have zero at z = 0, Infact it doesnt have any zeros In conclusion "no zeros exist outside the Region of convergence"
Reply by Martin Eisenberg December 26, 20062006-12-26
PrasadBC(CISC Tech) wrote:

> X(z) = 1 + az^-1+(az^-1)^2 +(az^-1)^3+.....+infinity; > .....(eq 1)
Writing "+ infinity" to indicate infinitely many terms mixes up two independent notions; don't do that.
> X(z) = 1/(1-az^-1); provided |az^-1|<1 => |z| >|a| > ......(eq 2) > > now X(z) can be written as X(z) = z/(z-a). > ......(eq 3) > > Now the question is what are the poles and zeros location of > X(z)???????? > > DSP Books says Zero at z = 0 Pole at z = a from (eq 3) > > but how can z=0 be substituted in the equation for X(z) = > 1/(1-az^-1) as 0 < |a| which is not in ROC and it makes the > original X(z) calculated using (eq 1) to go to infinity.
No. Each term in (1) is *indeterminate*, not infinite, at z = 0. Algebraic manipulation without regard to possible singularities -- such as augmenting (2) with z to get (3) -- amounts to implicit analytic continuation, thereby defining the value at a removable singularity. Martin -- Quidquid latine scriptum est, altum videtur.
Reply by Oli Charlesworth December 26, 20062006-12-26
On Tue, 26 Dec 2006 14:41:57 -0000, Rune Allnor <allnor@tele.ntnu.no>  =

wrote:

> > Oli Charlesworth skrev: > >> Again, the Z-transform does not converge *anywhere* outside the ROC, =
=
>> i.e. >> in our example, it doesn't converge for any |z|<=3D|a| (except possib=
ly at
>> the singular values specified as the zeros; I'm not too sure on that =
=
>> one!). > > So what is your point? That anything that happens inside the ROC > has to be discarded? If that's the case, all one would need was the > outermost pair of poles, and whatever zeros might be present > in the ROC.
I'm not sure I had a point! (and I'm not sure what you mean by "discarde= d") However, to analyse stability of a causal system (causality requires tha= t = the ROC include infinity), it is true that all you need to know are the = = locations of the outermost poles, as this places a bound on the ROC (the= = ROC cannot by definition include poles). If this ROC doesn't include th= e = unit circle, then the system is unstable. This is the formal explanation for why poles *must* be inside the unit = circle for a causal system to be stable. -- = Oli
Reply by Rune Allnor December 26, 20062006-12-26
Oli Charlesworth skrev:

> Again, the Z-transform does not converge *anywhere* outside the ROC, i.e. > in our example, it doesn't converge for any |z|<=|a| (except possibly at > the singular values specified as the zeros; I'm not too sure on that one!).
So what is your point? That anything that happens inside the ROC has to be discarded? If that's the case, all one would need was the outermost pair of poles, and whatever zeros might be present in the ROC. Rune
Reply by Oli Charlesworth December 26, 20062006-12-26
On Tue, 26 Dec 2006 14:11:13 -0000, Rune Allnor <allnor@tele.ntnu.no>  =

wrote:

> > Oli Charlesworth skrev: >> On Tue, 26 Dec 2006 13:29:27 -0000, Rune Allnor <allnor@tele.ntnu.no>=
>> wrote: >> > >> > The function X(z) =3D z/(z-a) has exactly one point where it does >> > not converge. So pedantically speaking, I suppose the ROC >> > can be defined as "everywhere except z=3Da". >> >> I think you are confusing two things, "convergence", and =
>> "singularities". >> >> The function X(z) =3D z/(z-a) does indeed have a singularity at z=3Da=
, but =
>> it >> is not meaningful to talk about "convergence" of X(z). >> >> The term "convergence" (and therefore "ROC") can only apply to an =
>> infinite >> summation; in our case it is meaningful to talk about the convergence=
=
>> of: >> >> X(z) =3D SUM x[n].z^-n (Eq.7) >> >> which has a clearly-defined ROC. If we only consider values within t=
hat
>> ROC, it is valid to specify X(z) as: >> >> X(z) =3D z/(z-a) (Eq.8) >> >> However, (Eq.8) is *not* the same as (Eq.7) outside the ROC. >> >> Specifying a Z-transform on its own doesn't uniquely define the origi=
nal
>> sequence. However, specifying a Z-transform *and* the ROC does uniqu=
ely
>> specify x[n]. > > The Z transform is defined as the sum from negative infinite > to positive infinite. The Z transform (7) is well-behaved everywhere > except at z=3Da, for a suitable choise of summation limits.
This is untrue. The Z-transform of a^n.u[n] is: X(z) =3D SUM a^n.z^-n and is not well-behaved (in the sense of convergence) for *any* value of= = |z|<=3D|a|, not just z=3Da. (This is in contrast to the expression z/(z-a), which is not well-behave= d = only at z=3Da. Hence it is a different function, which is why we must = specify the ROC when discussing z/(z-a)).
> It is > guaranteed not to converge for z =3D a, regardless of summation > limits.
This, however, is true (with the proviso that we choose the upper limit = as = +inf; for any other upper limit, it is a finite sum, and will always = "converge").
> Since the ZT is linear, the ZT of one function can be separated > as the sum of ZTs two functions. Since the sign of the exponent n > is crucial for the discussion, we can make an ARBITRARY > choise to separate x[n] to one series for negative n and > another for non-negative n: > > ZT{x[n]} =3D ZT{x1[n]} + ZT{x2[n]} (1) > > where > > ZT{x1[n]}^{-} =3D sum_{-inf}^-1 x[n] z^-n (2) > ZT{x2[n]}^{+} =3D sum{0}^{+inf} x[n] z^-n (3) > > Exploiting the linearity of the ZT like this and employing a > divide-and-conquer strategy, one can see that it converges > everywhere except at the poles.
Again, the Z-transform does not converge *anywhere* outside the ROC, i.e= . = in our example, it doesn't converge for any |z|<=3D|a| (except possibly = at = the singular values specified as the zeros; I'm not too sure on that one= !). -- = Oli
Reply by Rune Allnor December 26, 20062006-12-26
Oli Charlesworth skrev:
> On Tue, 26 Dec 2006 13:29:27 -0000, Rune Allnor <allnor@tele.ntnu.no> > wrote: > > > > > Oli Charlesworth skrev: > > > >> That doesn't really address the question though, which was (as I > >> understand it) "How can convergent points exist outside the ROC?", > > > > If that's the OP's question, I didn't catch it. > > > >> as the > >> term "ROC" would imply "the set of values of z which cause X(z) to > >> converge". I guess the answer is that singularities can exist outside > >> the > >> ROC, at which convergence still occurs, i.e. the definition of ROC is > >> actually "a subset of the values of z which cause X(z) to converge". > > > > The function X(z) = z/(z-a) has exactly one point where it does > > not converge. So pedantically speaking, I suppose the ROC > > can be defined as "everywhere except z=a". > > I think you are confusing two things, "convergence", and "singularities". > > The function X(z) = z/(z-a) does indeed have a singularity at z=a, but it > is not meaningful to talk about "convergence" of X(z). > > The term "convergence" (and therefore "ROC") can only apply to an infinite > summation; in our case it is meaningful to talk about the convergence of: > > X(z) = SUM x[n].z^-n (Eq.7) > > which has a clearly-defined ROC. If we only consider values within that > ROC, it is valid to specify X(z) as: > > X(z) = z/(z-a) (Eq.8) > > However, (Eq.8) is *not* the same as (Eq.7) outside the ROC. > > Specifying a Z-transform on its own doesn't uniquely define the original > sequence. However, specifying a Z-transform *and* the ROC does uniquely > specify x[n].
The Z transform is defined as the sum from negative infinite to positive infinite. The Z transform (7) is well-behaved everywhere except at z=a, for a suitable choise of summation limits. It is guaranteed not to converge for z = a, regardless of summation limits. Since the ZT is linear, the ZT of one function can be separated as the sum of ZTs two functions. Since the sign of the exponent n is crucial for the discussion, we can make an ARBITRARY choise to separate x[n] to one series for negative n and another for non-negative n: ZT{x[n]} = ZT{x1[n]} + ZT{x2[n]} (1) where ZT{x1[n]}^{-} = sum_{-inf}^-1 x[n] z^-n (2) ZT{x2[n]}^{+} = sum{0}^{+inf} x[n] z^-n (3) Exploiting the linearity of the ZT like this and employing a divide-and-conquer strategy, one can see that it converges everywhere except at the poles. The convention is to define as the ROC the region where the ZT converges under the summation limits as in (3). This choise is CONVENIENT since we usually want to discuss causal sequences. Rune
Reply by Oli Charlesworth December 26, 20062006-12-26
On Tue, 26 Dec 2006 13:29:27 -0000, Rune Allnor <allnor@tele.ntnu.no>  =

wrote:

> > Oli Charlesworth skrev: > >> That doesn't really address the question though, which was (as I >> understand it) "How can convergent points exist outside the ROC?", > > If that's the OP's question, I didn't catch it. > >> as the >> term "ROC" would imply "the set of values of z which cause X(z) to >> converge". I guess the answer is that singularities can exist outsid=
e =
>> the >> ROC, at which convergence still occurs, i.e. the definition of ROC is=
>> actually "a subset of the values of z which cause X(z) to converge". > > The function X(z) =3D z/(z-a) has exactly one point where it does > not converge. So pedantically speaking, I suppose the ROC > can be defined as "everywhere except z=3Da".
I think you are confusing two things, "convergence", and "singularities"= . The function X(z) =3D z/(z-a) does indeed have a singularity at z=3Da, b= ut it = is not meaningful to talk about "convergence" of X(z). The term "convergence" (and therefore "ROC") can only apply to an infini= te = summation; in our case it is meaningful to talk about the convergence of= : X(z) =3D SUM x[n].z^-n (Eq.7) which has a clearly-defined ROC. If we only consider values within that= = ROC, it is valid to specify X(z) as: X(z) =3D z/(z-a) (Eq.8) However, (Eq.8) is *not* the same as (Eq.7) outside the ROC. Specifying a Z-transform on its own doesn't uniquely define the original= = sequence. However, specifying a Z-transform *and* the ROC does uniquely= = specify x[n].
> I think the usual terminology in context of the z transform is > that the ROC denotes a region 0 <=3D b < |z| < c <=3D inf where > no poles exist. > >> In summary, I'm thoroughly confused! > > Me too. After posting my first response, I thought along these > lines: > > Let P be a polynomial in x, > > P(x) =3D a_n x^n + ... + a_1 x + 1 > > Multiply by x^-n/x^-n =3D 1 and re-order coefficients: > > P'(x) =3D x^-n +a_1 x^(n-1) + ... + a_n > > Since we only multiplied by 1, P'(x)=3DP(x). > > However, P'(0) =3D infinite because of the 1/x terms, > while P(0) =3D 1. > > Where is the flaw?
The flaw is that you multiplied by x^-n, not (x^-n/x^-n)! -- = Oli