dsp & digital filters
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DSP & Digital Filters
Lecture 1 z-Transform
DR TANIA STATHAKIREADER (ASSOCIATE PROFESSOR) IN SIGNAL PROCESSINGIMPERIAL COLLEGE LONDON
โข Recall that in order to describe a continuous-time signal ๐ฅ(๐ก) in frequency
domain we use:
โ The Continuous-Time Fourier Transform (or Fourier Transform):
๐(๐) = เถฑ
โโ
โ
๐ฅ(๐ก)๐โ๐๐๐ก๐๐ก
โ The Laplace Transform
๐ ๐ = เถฑ
โโ
โ
๐ฅ(๐ก)๐โ๐ ๐ก๐๐ก
โข The above transforms and their basic properties are considered known in
this course.
โข If you have doubts please consult any book on Signals and Systems.
Continuous-time signals
โข Consider a discrete-time signal ๐ฅ(๐ก) sampled every ๐ seconds.
๐ฅ ๐ก = ๐ฅ0๐ฟ ๐ก + ๐ฅ1๐ฟ ๐ก โ ๐ + ๐ฅ2๐ฟ ๐ก โ 2๐ + ๐ฅ3๐ฟ ๐ก โ 3๐ +โฏ
โข Recall that in the Laplace domain we have:
โ ๐ฟ ๐ก = 1โ ๐ฟ ๐ก โ ๐ = ๐โ๐ ๐
โข Therefore, the Laplace transform of ๐ฅ ๐ก is:
๐ ๐ = ๐ฅ0 + ๐ฅ1๐โ๐ ๐ + ๐ฅ2๐
โ๐ 2๐ + ๐ฅ3๐โ๐ 3๐ +โฏ
โข Now define ๐ง = ๐๐ ๐ = ๐(๐+๐๐)๐ = ๐๐๐cos๐๐ + ๐๐๐๐sin๐๐.
โข Finally, define
๐[๐ง] = ๐ฅ0 + ๐ฅ1๐งโ1 + ๐ฅ2๐ง
โ2 + ๐ฅ3๐งโ3 +โฏ
Discrete-time signals
The z-transform derived from the Laplace transform
โข From the Laplace time-shift property, we know that an additional term
๐ง = ๐๐ ๐ in the Laplace domain, corresponds to time-advance by ๐seconds (๐ is the sampling period) of the original function in time.
โข Accordingly, ๐งโ1 = ๐โ๐ ๐ corresponds to a time-delay of one sampling
period.
โข As a result, all sampled data (and discrete-time systems) can be
expressed in terms of the variable ๐ง.
โข More formally, the unilateral ๐ โ transform of a causal sampled
sequence:
๐ฅ ๐ = {๐ฅ 0 , ๐ฅ 1 , ๐ฅ 2 , ๐ฅ 3 , โฆ }
is given by:
๐[๐ง] = ๐ฅ0 + ๐ฅ1๐งโ1 + ๐ฅ2๐ง
โ2 + ๐ฅ3๐งโ3 +โฏ = ฯ๐=0
โ ๐ฅ[๐]๐งโ๐, ๐ฅ๐ = ๐ฅ[๐]
โข The bilateral ๐ โtransform for any sampled sequence is:
๐[๐ง] =
๐=โโ
โ
๐ฅ[๐]๐งโ๐
๐โ๐: the sampling period delay operator
โข Find the ๐ง โtransform of the causal signal ๐พ๐๐ข[๐], where ๐พ is a constant.
โข By definition:
๐[๐ง] =
๐=โโ
โ
๐พ๐๐ข ๐ ๐งโ๐ =
๐=0
โ
๐พ๐๐งโ๐ =
๐=0
โ๐พ
๐ง
๐
= 1 +๐พ
๐ง+
๐พ
๐ง
2
+๐พ
๐ง
3
+โฏ
โข We apply the geometric progression formula:
1 + ๐ฅ + ๐ฅ2 + ๐ฅ3 +โฏ =1
1 โ ๐ฅ, ๐ฅ < 1
โข Therefore,
๐[๐ง] =1
1โ๐พ
๐ง
, ๐พ
๐ง< 1
=๐ง
๐งโ๐พ, ๐ง > ๐พ
โข We notice that the ๐ง โtransform exists for certain values of ๐ง. These values
form the so called Region-of-Convergence (ROC) of the transform.
Example: Find the ๐ โtransform of ๐ ๐ = ๐ธ๐๐[๐]
โข Observe that a simple rational equation in ๐ง-domain corresponds to an
infinite sequence of samples in time-domain.
โข The figures below depict the signal in time (left) for ๐พ < 1 and the ROC,
shown with the shaded area, within the ๐ง โplane.
Example: Find the ๐ โtransform of ๐ ๐ = ๐ธ๐๐ ๐ cont.
โข Consider the causal signal ๐ฅ ๐ = ฯ๐=1๐พ ๐พ๐
๐๐ข[๐] with ๐(๐ง) = ฯ๐=1๐พ ๐ง
๐งโ๐พ๐.
โข In that case the ROC is the intersection of the ROCs of the individual
terms, i.e., the intersection of the sets ๐ง > ๐พ๐ i.e., ROC: ๐ง > ๐พmax
โข In case that ๐ฅ ๐ is the impulse response of a system, the transfer function
of the system is the rational function ๐(๐ง) = ฯ๐=1๐พ ๐ง
๐งโ๐พ๐with poles ๐พ๐.
โข The above analysis yields the following properties regarding the ROC:
PROPERTY:
If ๐ ๐ is a causal signal, the ROC of its ๐ โtransform is ๐ > ๐ธ๐ฆ๐๐ฑ
with ๐ธ๐ฆ๐๐ฑ the maximum magnitude pole of the ๐ โtransform.
โ In the general case of ๐ ๐ being a right-sided signal (RSS) the
ROC is as above but might not include โ (think why).
PROPERTY:
No pole can exist in ROC.
Generic form of a causal signal
โข The signal ๐ฅ ๐ = ฯ๐=1๐พ ๐พ๐
๐๐ข[๐] is bounded only if ๐พ๐ < 1 โ๐ or ๐พmax < 1.
โข In that case the ROC includes a circle with radius equal to 1. This is known
as the unit circle.
โข The above observation yields the following property:
PROPERTY:
If the ROC of ๐ฟ(๐) includes the unit circle in ๐ โplane, then the signal
in time is bounded and its Discrete Time Fourier Transform exists.
โข In case that ๐พ๐๐ข ๐ is part of a causal systemโs impulse response, we see
that the condition ๐พ < 1 must hold. This is because, since lim๐โโ
๐พ ๐ = โ,
for ๐พ > 1, the system will be unstable in that case.
โข Therefore, in causal systems, stability requires that the ROC of the
systemโs transfer function includes the unit circle.
Generic form of a causal signal cont.
โข Find the ๐ง โtransform of the anti-causal signal โ๐พ๐๐ข[โ๐ โ 1], where ๐พ is a
constant.
โข By definition:
๐[๐ง] =
๐=โโ
โ
โ๐พ๐๐ข โ๐ โ 1 ๐งโ๐ =
๐=โโ
โ1
โ๐พ๐๐งโ๐ = โ
๐=1
โ
๐พโ๐๐ง๐ = โ
๐=1
โ๐ง
๐พ
๐
= โ๐ง
๐พ
๐=0
โ๐ง
๐พ
๐
= โ๐ง
๐พ1 +
๐ง
๐พ+
๐ง
๐พ
2
+๐ง
๐พ
3
+โฏ
โข Therefore,
๐[๐ง] = โ๐ง
๐พ
1
1โ๐ง
๐พ
, ๐ง
๐พ< 1
=๐ง
๐งโ๐พ, ๐ง < ๐พ
โข We notice that the ๐ง โtransform exists for certain values of ๐ง, which consist
the complement of the ROC of the function ๐พ๐๐ข[๐] with respect to the
๐ง โplane.
Example: Find the ๐ โtransform of ๐ ๐ = โ๐ธ๐๐[โ๐ โ ๐]
โข Consider the anti-causal signal ๐ฅ ๐ = ฯ๐=1๐พ โ๐พ๐
๐๐ข[โ๐ โ 1] with
๐ง โtransform ๐(๐ง) = ฯ๐=1๐พ ๐ง
๐งโ๐พ๐.
โข In that case the ROC is the intersection of the sets ๐ง < ๐พ๐ , i.e., ROC:
๐ง < ๐พmin
โข In case that ๐ฅ ๐ is the impulse response of a system, the transfer function
of the system is the rational function ๐(๐ง) = ฯ๐=1๐พ ๐ง
๐งโ๐พ๐with poles ๐พ๐.
โข The above analysis yield the following property regarding ROCs:
PROPERTY:
If ๐ ๐ is an anti-causal signal, the ROC of its ๐ โtransform is ๐ <๐พmin with ๐พmin the minimum magnitude pole of the ๐ โtransform.
โ In the general case of ๐ ๐ being a left-sided signal (LSS) the ROC
is as above but might not include ๐ (think why).
Generic form of an anti-causal signal
โข We proved that the following two functions:
โช The causal function ๐พ๐๐ข ๐ and
โช the anti-causal function โ๐พ๐๐ข[โ๐ โ 1] have:
โ The same analytical expression for their ๐ง โtransforms.
โ Complementary ROCs. More specifically, the union of their ROCS
forms the entire ๐ง โplane.
โข The above observations verify that the analytical expression alone is not
sufficient to define the ๐ง โtransform of a signal. The ROC is also required.
Summary of previous examples
โข Example: Find the ๐ง โtransform of the two-sided signal:
๐ฅ ๐ = 2๐๐ข[๐] โ 4๐๐ข[โ๐ โ 1]
Based on the previous analysis we have:
๐ ๐ง =๐ง
๐งโ2+
๐ง
๐งโ4, ROC: ๐ง > 2 โฉ ๐ง < 4 or ROC: 2 < ๐ง < 4
โข Example: Find the ๐ง โtransform of the two-sided signal:
๐ฅ ๐ = 4๐๐ข[๐] โ 2๐๐ข[โ๐ โ 1]
Based on the previous analysis we have:
๐ ๐ง =๐ง
๐งโ2+
๐ง
๐งโ4, ROC: ๐ง > 4 โฉ ๐ง < 2 or ROC: โ
PROPERTY:
If ๐ ๐ is two-sided signal then the ROC of its ๐ โtransform is of the
form:
โ ๐พ1 < ๐ < ๐พ2 with ๐พ1, ๐พ2 poles of the system or
โ โ
Two-sided signals
โข By definition ๐ฟ 0 = 1 and ๐ฟ ๐ = 0 for ๐ โ 0.
๐[๐ง] =
๐=โโ
โ
๐ฟ ๐ ๐งโ๐ = ๐ฟ 0 ๐งโ0 = 1
โข By definition ๐ข ๐ = 1 for ๐ โฅ 0.
๐[๐ง] = ฯ๐=โโโ ๐ข ๐ ๐งโ๐ = ฯ๐=0
โ ๐งโ๐ =1
1โ1
๐ง
, 1
๐ง< 1
=๐ง
๐งโ1, ๐ง > 1
Example: Find the ๐ โtransform of ๐น[๐] and ๐[๐]
โข We write cos๐ฝ๐ =1
2๐๐๐ฝ๐ + ๐โ๐๐ฝ๐ .
โข From previous analysis we showed that:
๐พ๐๐ข[๐] โ๐ง
๐งโ๐พ, ๐ง > ๐พ
โข Hence,
๐ยฑ๐๐ฝ๐๐ข[๐] โ๐ง
๐งโ๐ยฑ๐๐ฝ, ๐ง > ๐ยฑ๐๐ฝ = 1
โข Therefore,
๐ ๐ง =1
2
๐ง
๐งโ๐๐๐ฝ+
๐ง
๐งโ๐โ๐๐ฝ=
๐ง(๐งโcos๐ฝ)
๐ง2โ2๐งcos๐ฝ+1, ๐ง > 1
Example: Find the ๐ โtransform of ๐๐จ๐ฌ๐ท๐๐[๐]
โข Find the ๐ง โtransform of the signal depicted in the figure.
โข By definition:
๐ ๐ง = 1 +1
๐ง+1`
๐ง2+1
๐ง3+1
๐ง4=
๐=0
4
๐งโ1 ๐ =1 โ ๐งโ1 5
1 โ ๐งโ1=
๐ง
๐ง โ 11 โ ๐งโ5
๐ โtransform of 5 impulses
Inverse ๐ โtransform
โข As with other transforms, inverse ๐ง โtransform is used to derive ๐ฅ[๐]from ๐[๐ง], and is formally defined as:
๐ฅ ๐ =1
2๐๐เถป๐[๐ง]๐ง๐โ1๐๐ง
โข Here the symbol ืฏ indicates an integration in counter-clockwise
direction around a circle within the ROC and ๐ง = ๐ ๐๐๐.
โข Such contour integral is difficult to evaluate (but could be done using
Cauchyโs residue theorem), therefore we often use other techniques to
obtain the inverse ๐ง โtransform.
โข One such technique is to use a ๐ง โtransform pairs Table shown in the
last two slides with partial fraction expansion.
Inverse ๐ โtransform: Proof
Proof:
1
2๐๐เถป๐[๐ง]๐ง๐โ1๐๐ง =
1
2๐๐เถป
๐=โโ
โ
๐ฅ[๐]๐งโ๐ ๐ง๐โ1๐๐ง
= ฯ๐=โโโ ๐ฅ[๐]
1
2๐๐ืฏ ๐ง๐โ๐โ1๐๐ง = ฯ๐=โโ
โ ๐ฅ[๐] ๐ฟ ๐ โ๐ = ๐ฅ[๐]
โ For the above we used the Cauchyโs theorem:
1
2๐๐ืฏ ๐ง๐โ1๐๐ง = ๐ฟ ๐ for ๐ง = ๐ ๐๐๐ anti-clockwise.
๐๐ง
๐๐= ๐๐ ๐๐๐ โ
1
2๐๐ืฏ ๐ง๐โ1๐๐ง =
1
2๐๐๐=02๐
๐ ๐โ1๐๐(๐โ1)๐ ๐๐ ๐๐๐๐๐ =
๐ ๐
2๐๐=02๐
๐๐๐๐ ๐๐ = ๐ ๐ ๐ฟ ๐
[ ๐ ๐
2๐๐=02๐
๐๐๐๐ ๐๐ = แ0 ๐ โ 0
๐ ๐
2๐2๐ = ๐ ๐ ๐ = 0
]
Find the inverse ๐ โtransform in the case of real unique poles
โข Find the inverse ๐ง โtransform of ๐ ๐ง =8๐งโ19
(๐งโ2)(๐โ3)
Solution
๐[๐ง]
๐ง=
8๐ง โ 19
๐ง(๐ง โ 2)(๐ โ 3)=(โ
196)
๐ง+
3/2
๐ง โ 2+
5/3
๐ง โ 3
๐ ๐ง = โ19
6+
3
2
๐ง
๐งโ2+
5
3
๐ง
๐งโ3
By using the simple transforms that we derived previously we get:
๐ฅ ๐ = โ19
6๐ฟ[๐] +
3
22๐ +
5
33๐ ๐ข[๐]
Find the inverse ๐ โtransform in the case of real repeated poles
โข Find the inverse ๐ง โtransform of ๐ ๐ง =๐ง(2๐ง2โ11๐ง+12)
(๐งโ1)(๐งโ2)3
Solution
๐[๐ง]
๐ง=
(2๐ง2โ11๐ง+12)
(๐งโ1)(๐งโ2)3=
๐
๐งโ1+
๐0
(๐งโ2)3+
๐1
(๐งโ2)2+
๐2
(๐งโ2)
โช We use the so called covering method to find ๐ and ๐0
๐ = เธญ(2๐ง2 โ 11๐ง + 12)
(๐ง โ 1)(๐ง โ 2)3๐ง=1
= โ3
๐0 = เธญ(2๐ง2 โ 11๐ง + 12)
(๐ง โ 1)(๐ง โ 2)3๐ง=2
= โ2
The shaded areas above indicate that they are excluded from the entire
function when the specific value of ๐ง is applied.
Find the inverse ๐ โtransform in the case of real repeated poles cont.
โข Find the inverse ๐ง โtransform of ๐ ๐ง =๐ง(2๐ง2โ11๐ง+12)
(๐งโ1)(๐งโ2)3
Solution
๐[๐ง]
๐ง=
(2๐ง2โ11๐ง+12)
(๐งโ1)(๐งโ2)3=
โ3
๐งโ1+
โ2
(๐งโ2)3+
๐1
(๐งโ2)2+
๐2
(๐งโ2)
โช To find ๐2 we multiply both sides of the above equation with ๐ง and let
๐ง โ โ.
0 = โ3 โ 0 + 0 + ๐2 โ ๐2 = 3
โช To find ๐1 let ๐ง โ 0.
12
8= 3 +
1
4+
๐1
4โ
3
2โ ๐1 = โ1
๐[๐ง]
๐ง=
(2๐ง2โ11๐ง+12)
(๐งโ1)(๐งโ2)3=
โ3
๐งโ1โ
2
๐งโ2 3 โ1
(๐งโ2)2+
3
(๐งโ2)โ
๐ ๐ง =โ3๐ง
๐งโ1โ
2๐ง
๐งโ2 3 โ๐ง
(๐งโ2)2+
3๐ง
(๐งโ2)
Find the inverse ๐ โtransform in the case of real repeated poles cont.
๐ ๐ง =โ3๐ง
๐งโ1โ
2๐ง
๐งโ2 3 โ๐ง
(๐งโ2)2+
3๐ง
(๐งโ2)
โข We use the following properties:
โช ๐พ๐๐ข[๐] โ๐ง
๐งโ๐พ
โช๐(๐โ1)(๐โ2)โฆ(๐โ๐+1)
๐พ๐๐!๐พ๐๐ข ๐ โ
๐ง
(๐งโ๐พ)๐+1
[โ2๐ง
๐งโ2 3 = (โ2)๐ง
๐งโ2 2+1 โ (โ2)๐(๐โ1)
222!๐พ๐๐ข ๐ = โ2
๐(๐โ1)
8โ 2๐๐ข[๐]
โข Therefore,
๐ฅ ๐ = [โ3 โ 1๐ โ 2๐(๐โ1)
8โ 2๐ โ
๐
2โ 2๐ + 3 โ 2๐]๐ข[๐]
= โ 3 +1
4๐2 + ๐ โ 12 2๐ ๐ข[๐]
Find the inverse ๐ โtransform in the case of complex poles
โข Find the inverse ๐ง โtransform of ๐ ๐ง =2๐ง(3๐ง+17)
(๐งโ1)(๐ง2โ6๐ง+25)
Solution
๐ ๐ง =2๐ง(3๐ง + 17)
(๐ง โ 1)(๐ง โ 3 โ ๐4)(๐ง โ 3 + ๐4)
๐[๐ง]
๐ง=
(2๐ง2โ11๐ง+12)
(๐งโ1)(๐งโ2)3=
๐
๐งโ1+
๐0
(๐งโ2)3+
๐1
(๐งโ2)2+
๐2
(๐งโ2)
Whenever we encounter a complex pole we need to use a special partial
fraction method called quadratic factors method.
๐[๐ง]
๐ง=
2(3๐ง+17)
(๐งโ1)(๐ง2โ6๐ง+25)=
2
๐งโ1+
๐ด๐ง+๐ต
๐ง2โ6๐ง+25
We multiply both sides with ๐ง and let ๐ง โ โ:
0 = 2 + ๐ด โ ๐ด = โ2
Therefore,
2(3๐ง+17)
(๐งโ1)(๐ง2โ6๐ง+25)=
2
๐งโ1+
โ2๐ง+๐ต
๐ง2โ6๐ง+25
Find the inverse ๐ โtransform in the case of complex poles cont.
2(3๐ง+17)
(๐งโ1)(๐ง2โ6๐ง+25)=
2
๐งโ1+
โ2๐ง+๐ต
๐ง2โ6๐ง+25
To find ๐ต we let ๐ง = 0:
โ34
25= โ2 +
๐ต
25โ ๐ต = 16
๐[๐ง]
๐ง=
2
๐งโ1+
โ2๐ง+16
๐ง2โ6๐ง+25โ ๐ ๐ง =
2๐ง
๐งโ1+
๐ง(โ2๐ง+16)
๐ง2โ6๐ง+25
โข We use the following property:
๐ ๐พ ๐ cos ๐ฝ๐ + ๐ ๐ข[๐] โ๐ง(๐ด๐ง+๐ต)
๐ง2+2๐๐ง+ ๐พ 2 with ๐ด = โ2, ๐ต = 16, ๐ = โ3, ๐พ = 5.
๐ =๐ด2 ๐พ 2+๐ต2โ2๐ด๐๐ต
๐พ 2โ๐2=
4โ25+256โ2โ(โ2)โ(โ3)โ16
25โ9= 3.2, ๐ฝ = cosโ1
โ๐
๐พ= 0.927๐๐๐,
๐ = tanโ1๐ด๐โ๐ต
๐ด ๐พ 2โ๐2= โ2.246๐๐๐.
Therefore, ๐ฅ ๐ = [2 + 3.2 cos 0.927๐ โ 2.246 ]๐ข[๐]
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