Transform Theory Notes

Keith M. Chugg c©September 22, 2015

Contents

1 Purpose and Scope 2

2 Fourier Family as Eigenvector Expansions 22.1 Discrete Fourier Transform T =ZN . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2.1.1 Relation Between the Normalized DFT and Circulant Matrix Theory . . . . 52.2 Fourier Series T =RT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Discrete Time Fourier Transform T =Z . . . . . . . . . . . . . . . . . . . . . . . . . . 82.4 Fourier Transform T =R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3 Selected Properties and Transform Pairs 12

4 Relation Between Fourier Family Members 244.1 Band Limited Signals - DTFT and Time Sampling . . . . . . . . . . . . . . . . . . . 244.2 Time Limited Signals - FS and Frequency Sampling . . . . . . . . . . . . . . . . . . 264.3 Approximately Time and Band Limited Signals - DFT . . . . . . . . . . . . . . . . . 284.4 The Time and Bandlimited Paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

5 Laplace and Z Transforms 305.1 Z Transform of a Stable, Causal System . . . . . . . . . . . . . . . . . . . . . . . . . 325.2 Laplace Transform of a Stable, Causal System . . . . . . . . . . . . . . . . . . . . . . 32

1 Purpose and Scope

These notes are intended to supplement the general theory of linear time invariant systems de-veloped in the “EE562a Supplemental Class Notes” by Prof. Scholtz. They provide a review oftransform theory with emphasis on the material relevant to EE562a. As such these notes do notprovide a comprehensive review EE401 or the equivalent.

The purpose of these notes is to relate the general theory from the supplemental notes to notionsthat you should already be familiar with. Specifically, the following are covered:

• The “Fourier Family” of transforms. These are the Fourier Transform (FT), Fourier Series(in time) (FS), Discrete Time Fourier Transform (DTFT) and the Discrete Fourier Transform(DFT).

– How these transforms correspond to orthogonal expansions in terms of the eigenvectorsof LTI systems for each of the four time index sets discussed in the supplemental notes.

– A consistent treatment of all four of these transforms with linear frequency.

– A description of the relationship between the four above transforms and the types ofreal world signals to which each applies.

– Tables of properties and important transform pairs.

• The Z-transform and Laplace Transform will be discussed briefly with emphasis placed ontheir relationship to Fourier family. Determination of the causality of the system from the Zor Laplace transform will also be discussed.

2 Fourier Family as Eigenvector Expansions

Motivation for using the FT, FS, DTFT and DFT will be given in this section. Four time sets willbe considered and in each case the corresponding transform will be developed as an expansion ofa given signal in terms of the complete orthonormal basis of eigenvectors of linear time invariant(LTI) systems on the time set. The index sets are considered in the order of least mathematicalcomplication, so that when sticky points are encountered in the more complex cases, we can appealto analogy with the mathematically simpler cases.

2.1 Discrete Fourier Transform T=ZNFor the index set T = ZN = {0, 1, . . . N − 1}, the space of signals, S, which are of interest arethose which are completely defined by N complex numbers x = {x[n] : n ∈ ZN}. Here we use thenotation of x ∈ S to represent a point in the abstract signal space.

In addition to these N complex numbers there may be some side information maintained aboutS. An example of such side information is that x actually represents a discrete time signal, x[n],which is defined for all integer values of n (i.e. not just n ∈ ZN ). When this interpretation is taken,the behavior of x[n] for n /∈ ZN is fully described by this side information. Common examples arethat x[n] = 0 ∀ n /∈ ZN and x[n] is periodic in n with period N . It will become apparent thatit is most natural to always think of x[n] as periodic, and this is the convention followed in thesenotes. In any case, the bottom line is that the behavior of x[n] is described fully by its values forn ∈ ZN plus some side information. While we treat x[n] as periodic, any other side informationcan be imposed after the fact by using the values of x[n] for n ∈ ZN .

2

Any linear mapping, H, of one complex signal, x ∈ S, into another, y ∈ S, defined on this indexset, so that y = Hx, may be represented as a finite superposition sum

y[n1] =

N−1∑

n2=0

h[n1, n2]x[n2]. (1)

The system response observed at time n1 to a Kronecker delta applied at time n2 is h[n1, n2].When the system is time invariant on this index set, the response at time n1 to an input at timen2 depends only the difference between n1 and n2 in the subtraction operator for this index set.

This implies that h[n1, n2] = h[n1�Nn2]; i.e. h is only a function of m = n1�Nn2∆= (n1−n2) mod

N, so that the superposition sum in (1) becomes a discrete time circular convolution

y[n1] = x[n1]~Nh[n1]∆=

N−1∑

n2=0

h[n1�Nn2]x[n2]. (2)

If the input to a LTI system is x = ek = {ek[n] = 1√N

exp(j 2πN kn

): n ∈ ZN}, where k is any

integer, then the output will be a scaled version of this same signal

y[n1] =N−1∑

n2=0

h[n1�Nn2]ek[n2] (3)

=N−1∑

n2=0

h[n2]ek[n1�Nn2] (4)

=1√N

N−1∑

n2=0

h[n2]ej2πNk(n1�Nn2) (5)

=

(N−1∑

n2=0

h[n2]e−j2πNkn2

)1√Nej

2πNkn1 (6)

= H[k]ek[n1]. (7)

Thus y = Hek = H[k]ek, so ek is an eigenvector of the system corresponding eigenvalue H[k],defined as

H[k] =N−1∑

n2=0

h[n2]e−j2πNkn2 . (8)

Since it is so easy to describe the output of a LTI system when the input is ek the summationin (2) can be simplified if we can express the input and output of the system in terms of theseeigenvectors. Also, the fact that ek = ek+rN for any integer r, implies that there are only N uniqueeigenvectors for LTI systems on this index set. The proper inner product for this signal space is

〈x,y〉 ∆=

N−1∑

n=0

x[n]y∗[n]. (9)

3

This implies that the N eigenvectors are orthonormal since1

〈ek, el〉 =

N−1∑

n=0

ek[n]e∗l [n] =1

N

N−1∑

n=0

ej2πN

(k−l)n = δK [k − l] =

{0 k 6= l1 k = l.

(10)

For the time set ZN , the dimension of the S is N . This implies that the set E = {ek : k ∈ ZN} isa complete orthonormal basis for the signal space. This means that any signal can be expressed asa linear combination of these basis vectors

x =

N−1∑

l=0

X[l]el. (11)

The coefficients, X[k], can be found by taking the inner product of both sides with ek

〈x, ek〉 =⟨∑N−1

l=0 X[l]el, ek

⟩(12)

=N−1∑

l=0

X[l]〈el, ek〉 (13)

=

N−1∑

l=0

X[l]δK [k − l] (14)

= X[k]. (15)

The expansion in (11) can be written as N expansion equations and N coefficient equations(referred to as the “Normalized DFT2”)

x[n] =

N−1∑

k=0

X[k]ek[n] =1√N

N−1∑

k=0

X[k]ej2πNkn n = 0, 1, . . . N − 1 (16a)

X[k] =N−1∑

n=0

x[n]e∗k[n] =1√N

N−1∑

n=0

x[n]e−j2πNkn k = 0, 1, . . . N − 1. (16b)

Note that this definition is only slightly different than the system eigenvalue, H[k], in (7). In fact itis clear that the kth eigenvalue of the system, H[k], is related to the kth coefficient of the eigenvectorexpansion of the impulse response h[n], H[k] by

H[k] =1√NH[k]. (17)

The eigenvector expansion of the system output is now easily obtained

y = Hx = HN−1∑

k=0

X[k]ek =

N−1∑

k=0

X[k]Hek =

N−1∑

k=0

X[k]H[k]ek, (18)

1This follows from the incomplete geometric sum

N−1∑n=0

αn =1− αN

1− α .

2This is also sometimes referred to as the Discrete Time Fourier Series for reasons that will become obvious inSection 4

4

thusY [k] = X[k]H[k] =

√NX[k]H[k]. (19)

The expansion and coefficient equations in (16a) and (16b) have all the essential qualities thatare associated with the DFT, but the conventional DFT is a slightly different. Instead of theorthonormal basis {ek}N−1

k=0 it is conventional to use the orthogonal basis {vk = ek/√N}N−1

k=0 forthe DFT definition. The only change that this convention introduces is a scale factor:

Expansion: x[n] = DFT−1 {X[k]} ∆=

1

N

N−1∑

k=0

X[k]ej2πNkn (20a)

Coefficients: X[k] = DFT {x[n]} ∆=

N−1∑

n=0

x[n]e−j2πNkn (20b)

The benifit of working with this basis is that the kth eigenvalue of the system, H[k], is alsothe kth DFT coefficient. We also have that (circular) convolution in the time domain impliesmultiplication in the DFT frequency domain (without the factor of

√N in (19))

Y [k] = H[k]X[k]. (21)

Notice that the expression for the Inverse-DFT of X[k] is periodic in n with period N . This iswhy it is most convenient to think of x[n] as periodic, otherwise a disclaimer specifying that theequality only holds for n ∈ ZN must be carried. Also noteworthy is the fact that structure of thetime and frequency domains are identical. In other words X[k] is defined by N complex numbersand can be interpreted as periodic in k with period N .

2.1.1 Relation Between the Normalized DFT and Circulant Matrix Theory

The signal space for the index set T = ZN really just corresponds to CN with all the fancy notationbeing used to stress the similarity with other index sets.3 The superposition sum in (1) is just amatrix multiplication and the circular convolution in (2) is a matrix multiplication with a circulantmatrix (check this!):

y[0]y[1]· · ·

y[N − 1]

︸ ︷︷ ︸y

=

h[0] h[N − 1] · · · h[1]h[1] h[0] · · · h[2]

......

. . ....

h[N − 1] h[N − 2] · · · h[0]

︸ ︷︷ ︸H

x[0]x[1]· · ·

x[N − 1]

.

︸ ︷︷ ︸x

(22)

The standard eigenvectors for the circulant matrix are

ek =1√N

exp(j 2πN k0

)

exp(j 2πN k1

)...

exp(j 2πN k(N − 1)

)

, (23)

3When we move to more complicated (infinite) index sets we cannot represent a linear system as a matrix, so the“signal view” is valuable for this reason also.

5

which is just ek[n] written in vector form. Using the standard matrix of eigenvectors

E =[

e0 e1 · · · eN−1

], (24)

allows the expansion and coefficient equations of the Normalized DFT in (16) to be written inmatrix form

Expansion: x = EX (25a)

Coefficient: X = E†x, (25b)

where the notation

X =

X[0]

X[1]· · ·

X[N − 1]

(26)

has been introduced and it has been noted that E is unitary as a result of the inner productdefinition in (9).

The system response is fully described by its eigenvalues since

y = Hx = HEX = EΛHX (27)

implies Y = ΛHX. The correspondence is completed by noting that

ΛH = diag(H[0], H[1], . . . H[N − 1]). (28)

This is just a vector version of (19). In other words, the eigenvalues of any circulant matrix are theDFT coefficients of the first column of the matrix. For a finite index set, we may either work with“signals” or vectors, but for LTI systems the matrix is defined by the first column, so the signalapproach is more compact.

2.2 Fourier Series T=RT

The Fourier Series (in time) is the eigenfunction expansion for the time set T = RT = [0, T ]. Thesignal space, S, in this case is all square integrable functions on RT .4 In other words x ∈ S meansthat x = {x(t) : t ∈ RT } where ∫ T

0|x(t)|2dt <∞. (29)

The inner product associated with this signal space is

〈x,y〉 ∆=

∫ T

0x(t)y∗(t)dt, (30)

so the requirement in (29) may be interpreted as requiring that signals have finite length. Weagain have the different possibilities of what x represents in reality. There may be side informationspecifying that x(t) = 0 ∀ t /∈ RT or that x(t) is periodic in t with period T . Just as in the

4The careful reader will notice that square integrability is not the condition for the Fourier Series to exist. Thisnecessary set of conditions, the Dirichlet conditions, are automatically satified when the signal is square integrable.The Dirichlet conditions require absolute integrability and other properties, we’ll use absolute integrability when wetalk about stability. Working with the smaller class of square integrable functions is nice for the development whichfollows the same track as the DFT development (it allows us to work in a Hilbert space).

6

DFT case, the distinction between these two possibilities is not a concern, but we will treat x(t) asperiodic.5

Any linear operator H on the index set T = RT can be represented as a superposition integral

y = Hx ⇐⇒ y(t1) =

∫ T

0h(t1, t2)x(t2)dt2, (31)

where h(t1, t2) is the system response at time t1 resulting from a Dirac delta applied at time t2. AnLTI system on this index set has the property that h(t1, t2) = h(t1�T t2), so that the superpositionsum in (31) becomes a continuous time circular convolution

y(t1) = x(t1)~Th(t1)∆=

∫ T

0h(t1�T t2)x(t2)dt2. (32)

An eigenvector for LTI systems in this case is ek = {ek(t) = 1√T

exp(j 2πT kt

): t ∈ RT },

where k is any integer. Taking the input to a LTI system, H, to be ek implies that the output isy = Hek =

√THkek, where Hk is given by

Hk =1√T

∫ T

0h(t)e−j

2πTktdt. (33)

This can be verified in the same manner used to derive (7).The description of the effect of the LTI system can be simplified if the signals can be expressed

as a linear combination of the eigenvectors. The eigenvectors are orthonormal since

〈ek, el〉 =

∫ T

0ek(t)e

∗l (t)dt =

1

T

∫ T

0ej

2πT

(k−l)tdt = δK [k − l]. (34)

The next question is: ‘how many of these eigenvectors do we need to form a complete orthonormalbasis for S?’ Since there are countably-infinite many ek’s, one would suspect that all of theseeigenvectors are required. If this is the case then the issue of convergence arises (i.e. the eigenvectorexpansions will be infinite).

Fortunately, the set E = {ek : k ∈ Z} is a complete orthonormal basis for the signal space.What this means in this case is that there is an expansion

x =∞∑

l=−∞Xlel, (35)

but the equality must be interpreted in the following sense

x = z ⇐⇒ 〈x− z,x− z〉 =

∫ T

0|x(t)− z(t)|2dt = 0. (36)

The physical interpretation is that there is no power (or energy as the case may be) in the differencebetween x and its eigenvector expansion.6

The coefficients can be evaluated as before

Xk = 〈x, ek〉 =1√T

∫ T

0x(t)e−j

2πTktdt. (37)

5The term in (29) is proportional to the average power if x(t) is periodic, or to the total energy if x(t) is zerooutside the interval [0, T ].

6The manifestation of this interpretation of the convergence is the so-called Gibbs phenomena.

7

This defines a “Normalized Fourier Series”

x(t) =

∞∑

k=−∞Xkek(t) =

1√T

∞∑

k=−∞Xke

j 2πTkt t ∈ [0, T ] (38a)

Xk =

∫ T

0x(t)e∗k(t)dt =

1√T

∫ T

0x(t)e−j

2πTkt k ∈ Z. (38b)

This definition is consistent with the notation of Hk for the eigenvalues in (33); the kth eigenvalueof the system,

√THk, is a scalar multiple of the kth coefficient of the eigenvector expansion of the

impulse response h(t). Using the same method as in the DFT discussion yields the output expansioncoefficients in terms of the input and system expansion coefficients

Yk =√TXkHk. (39)

Just as with the DFT, the convention in the signal processing literature is to use a slightlydifferent basis, namely {vk =

√Tek}∞k=−∞, which is orthogonal but not orthonormal. Once again,

the only change is a scale factor:

Expansion: x(t) = FS−1 {Xk} ∆=

∞∑

k=−∞Xke

j 2πTkt (40a)

Coefficients: Xk = FS {x(t)} ∆=

1

T

∫ T

0x(t)e−j

2πTkt (40b)

When working with the conventional FS, the frequency domain equivalent of convolution is

Yk = THkXk. (41)

Notice that the expression for the Inverse-FS of Xk is periodic in t with period T . This is why itis most natural to think of x(t) as periodic. Also notice that, in contrast to the DFT, the structureof the time and frequency domains are different; the FS {x(t)} is a countably infinite collection ofnumbers, while x represents uncountably many values.

2.3 Discrete Time Fourier Transform T=ZThe Discrete Time Fourier Transform is the eigensequence expansion for the time set T = Z ={. . . − 2,−1, 0, 1, 2, . . .}. The signal space, S, in this case is all square summable sequences. Inother words x ∈ S means that x = {xn : n ∈ Z} where

∞∑

n=−∞|xn|2 <∞. (42)

The inner product associated with this signal space is

〈x,y〉 ∆=

∞∑

n=−∞xny

∗n, (43)

The requirement in (42) can be considered a finite energy constraint on the signals.

8

Any linear operator BbbH on the index set T = Z can be represented by a superposition sum

yn1 =

∞∑

n2=−∞hn1,n2xn2 . (44)

which reduces to a discrete time convolution in the LTI case (i.e. hn1,n2 = hn1−n2)

yn1 = xn1∗hn1

∆=

∞∑

n2=−∞hn1−n2xn2 . (45)

A good guess for an LTI eigenvector for T = Z is eν = {en(ν) = exp(j2πνn) : n ∈ Z}, whereν is any real number. When the input to a LTI system, H, is eν , the output, y = Heν is

yn1 =∞∑

n2=−∞hn2en1−n2(ν) (46)

=

( ∞∑

n2=−∞hn2e

j2πνn2

)e−j2πνn1 (47)

= H(ν)en1(ν). (48)

So the eigenvalue associated with eν is H(ν).The above relation holds for any real number ν, but since eν = eν+r for any integer r we need

to only consider ν on an interval with length 1. The convergence problem which arose in the FScase, when there were a countable number of eigenvalues, is worsened for this index set since eν isan eigenvector for any real ν. It also turns out that in order to form a “basis” for S we need thefull set E = {eν : ν ∈ R1 = [0, 1]}. To further complicate matters these eigenvectors do not evenstrictly belong to the signal space! To see this consider the energy in eν

〈eν , eν〉 =∞∑

n=−∞|en(ν)|2 =

∞∑

n=−∞1 =∞. (49)

Things seems pretty hopeless at this point, however the theory of generalized functions solvesthe problem (for engineering purposes). Specifically, introduction of the Dirac delta function, whichis defined by the relation7

x(0) =

∫ b

ax(λ)δD(λ)dλ ∀ a < 0 < b, (x(t)continuous at t = 0) (50)

allows the concept of an orthonormal basis to be extended to an uncountable set. The generalizedorthonormal condition is then

〈eν1 , eν2〉 =

∞∑

n=−∞en(ν1)e∗n(ν2) =

∞∑

n=−∞ej2π(ν1−ν2)n = δD(ν1 − ν2) ν1, ν2 ∈ [0, 1]. (51)

The choice of [0,1] as the set for ν is of no particular significance since en(ν) is a periodic functionof ν with period 1. As in the prior cases, it is convenient to think of allowing ν to vary over all Rand treating en(ν) as periodic. With this interpretation the orthogonality condition becomes

〈eν1 , eν2〉 =

∞∑

r=−∞δD(ν1 − ν2 − r), (52)

7The Dirac delta is not really a function, but it is well defined if we think of it as a functional; that is it maps afunction, x(t), to a number, x(0). This is the type of reasoning behind a rigorous definition of the delta function.

9

which is valid even if ν1, ν2 /∈ [0, 1].To simplify the description of LTI systems, we express the signal as a linear combination of the

basis elements in Ex =

∫ 1

0X(ν)eνdν, (53)

where the integral replaces the sum used previously because of the continuous indexing of theeigenvectors. Again the equality must be interpreted as a zero energy error equivalence

x = z ⇐⇒ 〈x− z,x− z〉 =∞∑

n=−∞|xn − zn|2 = 0. (54)

The coefficients are found as usual, taking ν1 ∈ [0, 1]

〈x, eν1〉 =⟨∫ 1

0 X(ν2)eν2dν2, eν1

⟩(55)

=

∫ 1

0X(ν2)〈eν2 , eν1〉dν2 (56)

=

∫ 1

0X(ν2)δD(ν1 − ν2)dν2 (57)

= X(ν1). (58)

Expanding this expression using (43) and the expansion in (53) yields the DTFT (the normalizedversion is conventional in this case)

Expansion: xn = DTFT−1 {X(ν)} ∆=

∫ 1

0X(ν)ej2πνndν (59a)

Coefficients: X(ν) = DTFT {xn} ∆=

∞∑

n=−∞xne

−j2πνn (59b)

This definition is once again consistent with the notation used for the system eigenvalues; H(ν)is the eigenvalue associated with eν and the DTFT of the system impulse response. The timedomain system operation is also transferred to multiplication in the frequency domain

Y (ν) = H(ν)X(ν). (60)

The expression for the DTFT of xn is periodic in ν with period 1, hence the periodic inter-pretation of X(ν). The structures of the time and frequency domains for the DTFT are exactlyreversed from those in the FS. In fact the the transform in (59) can be viewed as a Fourier Seriesof X(ν) on R1 in the ν variable.

2.4 Fourier Transform T=RThe Continuous Time Fourier Transform, or simply the Fourier Transform, is the eigenfunctionexpansion for the time set of the real line, T = R. The signal space, S, of interest is all squareintegrable sequences. Thus x ∈ S means that x = {x(t) : t ∈ R} where

∫ ∞

−∞|x(t)|2dt <∞. (61)

10

The inner product associated with this signal space is

〈x,y〉 ∆=

∫ ∞

−∞x(t)y∗(t)dt, (62)

Once again, the requirement in (61) can be considered a finite energy constraint on the signals.Any linear operator H on index set T = R can be represented by a superposition integral

y(t1) =

∫ ∞

−∞h(t1, t2)x(t2)dt2. (63)

which reduces to a continuous time convolution in the LTI case (i.e. h(t1, t2) = h(t1 − t2))

y(t1) = x(t1)∗h(t1)∆=

∫ ∞

−∞h(t1 − t2)x(t2)dt2. (64)

At this point we know that an LTI eigenvector for T = R is ef = {e(t, f) = exp(ij2πft) : t ∈R}, where f is any real number. When the input to a LTI system, H, is ef , the output, y = Hefis

y(t1) =

∫ ∞

−∞h(t2)e(t1 − t2, f)dt2 (65)

=

(∫ ∞

−∞h(t2)e−ij2πft2dt2

)e(t1, f) (66)

= H(f)e(t1, f), (67)

therefore the eigenvalue associated with ef is H(f).Accepting the same complications encountered with the basis for the DTFT, we can take a

E = {ef : f ∈ R} as a complete orthonormal basis of eigenvectors. These functions are orthonormalin the same generalized sense introduced for the DTFT, namely

〈ef1 , ef2〉 =

∫ ∞

−∞e(t, f1)e∗(t, f2)dt =

∫ ∞

−∞eij2π(f1−f2)tdt = δD(f1 − f2). (68)

A signal is expressed as a linear combination of the basis elements in E

x =

∫ ∞

−∞X(f)efdf, (69)

with the standard interpretation

x = z ⇐⇒ 〈x− z,x− z〉 =

∫ ∞

−∞|x(t)− z(t)|2dt = 0. (70)

The coefficients are found as usualX(f) = 〈x, ef 〉. (71)

Expanding the abstract notation in (69)-(71) yields the Fourier Transform (again this normalizedversion is conventional)

Expansion: x(t) = FT−1 {X(f)} ∆=

∫ ∞

−∞X(f)eij2πftdf (72a)

Coefficients: X(f) = FT {x(t)} ∆=

∫ ∞

−∞x(t)e−ij2πftdt (72b)

11

Once again H(f) is the system eigenvalue as well as the Fourier Transform of the system impulseresponse and the output transform is

Y (f) = H(f)X(f). (73)

Like the DFT, the time and frequency domains associated with the Fourier Transform have thesame structure.

3 Selected Properties and Transform Pairs

Useful properties of the transforms developed in Section 2 are summarized in this section. Variationsof most of these properties can be found in most standard references, although some of the propertieswhich are useful for EE562a are a little obscure (one reason for writing these notes). The assumptionof periodicity is made in DFT table, the time domain of the FS table and the frequency domainof the DTFT table. In most cases other side information can be imposed (i.e. the function iszero outside the interval of interest), but there are some exceptions. Periodicity is necessary forthe summation properties of the DTFT and DFT and the integration property of the FS to bemeaningful. Also without the periodicity interpretation, many of the minus signs should be replacedby “modulo minus” signs.

The tables include some standard and not so standard notation. The notation used to denotethe Hermitian symmetric part and the Hermitian antisymmetric part is non standard, but simple

HS {x(λ)} ∆=x(λ) + x∗(−λ)

2(74)

HAS {x(λ)} ∆=x(λ)− x∗(−λ)

2. (75)

This is a direct extension of the even and odd parts of a real function. Notice that x(·) is Hermitiansymmetric, if and only if HS {x(λ)} = x(λ) and x(·) is Hermitian antisymmetric, if and only ifHAS {x(λ)} = x(λ). It is easier to state the symmetry properties of the transforms with thisconcept. This notion, extended to discrete time signals, along with the periodicity interpretation,especially simplifies the symmetry conditions of the DFT.

The other new notation introduced is for standard functions. The best practice is to pick aconvention for these functions and stick with it. My conventions are8

sgn(t) =

{1 t > 0−1 t < 0

(76)

u(t) = sgn(t)/2 + 1/2 =

{1 t > 00 t < 0

(77)

rect(t) =

{1 |t| < 1/20 |t| > 1/2

(78)

trian(t) =

{1− |t| |t| < 10 |t| > 1

(79)

sinc(t) =

{sin(πt)πt t 6= 0

1 t = 0, (80)

8At points of discontinuity it is best to take the function to be the average of the left and right limits (e.g.u(0) = 1/2). The reason is that the Fourier integral (for the FT) and Fourier sum (for the FS) converge to thisaverage value. It is not a major consideration for us in any event.

12

rect(t)

sgn( t) u(t)

trian( t)

+1

+1 +1

+1

0

0 0 1-11/2-1/2

-1

t

t

t

t

-4 -3 -2 -1 1 2 3 4

1

sinc( t)

t

0n

n

n

n

sgn[n] u[n]+1 +1

+12M+1

-1

M-M 2M-2M

DrectM[n] Dtrian M[n]

Transform Theory Notes 15

Figure 1: Continuous time functions

Figure 2: Discrete time functions

Figure 1: Continuous time functions

which are illustrated in Figure 1. Note that the value of the sinc function at zero is consistent withthe limit (use L’Hopitals rule).

Similar functions are of interest for discrete time. There are some modifications which must bemade due to the fact that the functions are only defined for integer values of the argument. Hereare the conventions used

sgn[n] =

{1 n ≥ 0−1 n < 0

(81)

u[n] = sgn[n]/2 + 1/2 =

{1 n ≥ 00 n < 0

(82)

DrectM [n] =

{1 |n| ≤M0 |n| > M

(83)

DtrianM [n] =

{2M + 1− |n| |n| ≤ 2M0 |n| > 2M

. (84)

The DrectM [n] and DtrianM [n] are discrete time versions of scaled rect and triangle functions (thisis nonstandard notation). These functions are sketched in Figure 2.

Another function which is usually not given a special name is also introduced

dincM (ν) =

{sin(πν(2M+1))(2M+1) sin(πν) ν 6= 0

1 ν = 0, (85)

where M is an integer and ν is any real number. Note that the value at ν = 0 is consistent with thelimiting value (i.e. it is continuous). We will see that in the discrete time interpolation formula, thisfunction plays a role similar to that of the sinc function in the continuous time sampling theorem,hence the name. Notice that the dinc function is periodic in ν with period 1 (regardless of thevalue of M). The dinc function is sketched for M = 2 in Figure 3.

13

rect(t)

sgn( t) u(t)

trian( t)

+1

+1 +1

+1

0

0 0 1-11/2-1/2

-1

t

t

t

t

-4 -3 -2 -1 1 2 3 4

1

sinc( t)

t

0n

n

n

n

sgn[n] u[n]+1 +1

+12M+1

-1

M-M 2M-2M

DrectM[n] Dtrian M[n]

Transform Theory Notes 15

Figure 1: Continuous time functions

Figure 2: Discrete time functionsFigure 2: Discrete time functions

-1.5 -1 -0.5 0.5 1 1.5

1

0ν

dinc 2(ν)

16 c⃝ K.M. Chugg - November 19, 1995

Figure 3: The dinc function

u(0) = 1/2). The reason is that the Fourier integral (for the FT) and Fourier sum (for the FS) converge tothis average value. It is not a major consideration for us in any event.

Figure 3: The dinc function

14

15

Fourier Transform Properties

FT Property Time Domain ←→ Frequency Domain

Convolution x(t)∗y(t) ←→ X(f)Y (f)

Linearity αx(t) + βy(t) ←→ αX(f) + βY (f)

Time Shift x(t− t0) ←→ e−j2πft0X(f)

Frequency Shift ej2πf0tx(t) ←→ X(f − f0)

Inner Product∫∞−∞ x(t)y∗(t)dt =

∫∞−∞X(f)Y ∗(f)df

Parseval’s Theorem∫∞−∞ |x(t)|2dt =

∫∞−∞ |X(f)|2df

Spectral Conjugation x∗(−t) ←→ X∗(f)

Autocorrelation x(t)∗x∗(−t) =∫∞−∞ x(λ)x∗(λ− t)dλ ←→ |X(f)|2

Modulation x(t)y(t) ←→ X(f)∗Y (f)

Scaling x(ct) c 6= 0 ←→ 1|c|X

(fc

)

Differentiation dx(t)/dt ←→ (j2πf)X(f)

Integration∫ t−∞ x(λ)dλ ←→ X(f)

(j2πf) + 12X(0)δD(f)

Real ∼ HS <{x(t)} ←→ HS {X(f)}

Imaginary ∼ HAS ={x(t)} ←→ −jHAS {X(f)}

HS ∼ Real HS {x(t)} ←→ <{X(f)}

HAS ∼ Imaginary HAS {x(t)} ←→ j={X(f)}

Duality X(t) ←→ x(−f)

16

Fourier Transform Pairs

No. Signal: x(t) Transform: X(f)

〈1〉 1 δD(f)

〈2〉 δD(t− t0) e−j2πft0

〈3〉 ej2πf0t δD(f − f0)

〈4〉 ∑∞n=−∞ δD(t− nTs) 1

Ts

∑∞n=−∞ δD(f − n/Ts)

〈5〉 cos(2πf0t)12(δD(f − f0) + δD(f + f0))

〈6〉 sin(2πf0t)12j (δD(f − f0)− δD(f + f0))

〈7〉 sgn(t) 1/(jπf)

〈8〉 u(t) 1/(j2πf) + 12δD(f)

〈9〉 rect(t) sinc(f)

〈10〉 trian(t) (sinc(f))2

〈11〉 e−πt2

e−πf2

〈12〉 e−a|t| a > 0 (real) 2a(2πf)2+a2

〈13〉 e−atu(t) <{a} > 0 1j2πf+a

〈14〉 1a−b

(e−bt − e−at

)u(t) <{a}, <{b} > 0 a 6= b 1

(j2πf+a)(j2πf+b)

〈15〉 e−at sin(2πf0t)u(t) <{a} > 0 2πf0[j2πf+(a+j2πf0)][j2πf+(a−j2πf0)]

〈16〉 te−atu(t) <{a} > 0 1(j2πf+a)2

〈17〉 tn−1

(n−1)!e−atu(t) <{a} > 0 1

(j2πf+a)n

17

Discrtete Time Fourier Transform Properties

DTFT Property Time Domain ←→ Periodic 〈1〉 Frequency Domain

Convolution xn∗yn ←→ X(ν)Y (ν)

Linearity αxn + βyn ←→ αX(ν) + βY (ν)

Time Shift xn−n0 ←→ e−j2πνn0X(ν)

Frequency Shift ej2πν0nxn ←→ X(ν − ν0)

Inner Product∑∞

n=−∞ xny∗n =

∫ 10 X(ν)Y ∗(ν)dν

Parseval’s Theorem∑∞

n=−∞ |x(t)|2dt =∫ 1

0 |X(ν)|2dν

Spectral Conjugation x∗−n ←→ X∗(ν)

Autocorrelation xn∗x∗−n ←→ |X(ν)|2

Modulation xnyn ←→ X(ν)~1Y (ν)

Scaling xrnr ∈ Z r 6= 0 ←→ 1|r|∑r−1

m=0X(ν+mr

)

Time Difference xn − xn−1 ←→ (1− e−j2πν)X(ν)

Summation∑n

m=−∞ xm ←→ (1− e−j2πν)−1X(ν) + 12X(0)

∑∞r=−∞ δD(ν − r)

Real ∼ HS <{xn} ←→ HS {X(ν)}

Imaginary ∼ HAS ={xn} ←→ −jHAS {X(ν)}

HS ∼ Real HS {xn} ←→ <{X(ν)}

HAS ∼ Imaginary HAS {xn} ←→ j={X(ν)}

Duality with FS FS {X(t/T )} = x−k

18

Discrete Time Fourier Transform Pairs

No. Signal: xn Periodic 〈1〉 Transform: X(ν)

〈1〉 1∑∞

m=−∞ δD(ν −m)

〈2〉 δK [n− n0] e−j2πνn0

〈3〉 ej2πν0n∑∞

m=−∞ δD(ν − ν0 −m)

〈4〉 ∑∞m=−∞ δK [n−mM ] 1

M

∑∞m=−∞ δD(ν −m/M)

〈5〉 cos(2πν0n) 12

∑∞m=−∞(δD(ν − ν0 −m) + δD(ν + ν0 −m))

〈6〉 sin(2πν0n) 12j

∑∞m=−∞(δD(ν − ν0 −m)− δD(ν + ν0 −m))

〈7〉 sgn[n] 21−e−j2πν

〈8〉 u[n] 11−e−j2πν + 1

2

∑∞m=−∞ δD(ν −m)

〈9〉 DrectM [n] (2M + 1)dincM (ν) = sin(2πν(M+1/2))sin(πν)

〈10〉 DtrianM [n] ((2M + 1)dincM (ν))2 =(

sin(2πν(M+1/2))sin(πν)

)2

〈11〉 2V sinc(2V n) 0 < V < 1/2∑∞

m=−∞ rect(ν−m2V

)

〈12〉 ρ|n| |ρ| < 1 (real) 1−ρ21+ρ2−2ρ cos(2πν)

〈13〉 anu[n] |a| < 1 11−ae−j2πν

〈14〉 1a−b

(bn+1 − an+1

)u[n] |a|, |b| < 1 a 6= b 1

(1−ae−j2πν)(1−be−j2πν)

〈15〉 ρn sin[2πν0(n+ 1)]u[n] |ρ| < 1 (real) sin(2πν0)(1−ρej2πν0e−j2πν)(1−ρe−j2πν0e−j2πν)

〈16〉 (n+ 1)anu[n] |a| < 1 1(1−ae−j2πν)2

〈17〉 (n+r−1)!n!(r−1)! a

nu[n] |a| < 1 1(1−ae−j2πν)r

19

Fourier Series Properties

FS Property Periodic (T ) Time Domain ←→ Frequency Domain

Convolution x(t)~T y(t) ←→ TXkYk

Linearity αx(t) + βy(t) ←→ αXk + βYk

Time Shift x(t− t0) ←→ e−j2πTkt0Xk

Frequency Shift ej2πTk0tx(t) ←→ Xk−k0

Inner Product 1T

∫ T0 x(t)y∗(t)dt =

∑∞k=−∞XkY

∗k

Parseval’s Theorem 1T

∫ T0 |x(t)|2dt =

∑∞k=−∞ |Xk|2

Spectral Conjugation x∗(−t) ←→ X∗k

Autocorrelation 1T x(t)~Tx

∗(−t) ←→ |Xk|2

Modulation x(t)y(t) ←→ Xk∗Yk

Scaling x(rt) r ∈ Z r 6= 0 ←→ sgn[r]∑∞

m=−∞Xmsinc(k −mr)

Differentiation dx(t)/dt ←→(j 2πT k)Xk

Integration∫ t−∞ x(λ)dλ ←→

(j 2πT k)−1

Xk if X0 = 0

Real ∼ HS <{x(t)} ←→ HS {Xk}

Imaginary ∼ HAS ={x(t)} ←→ −jHAS {Xk}

HS ∼ Real HS {x(t)} ←→ <{Xk}

HAS ∼ Imaginary HAS {x(t)} ←→ j={Xk}

Duality with DTFT DTFT {Xn} = x(−νT )

20

Fourier Series Pairs

No. Periodic (T ) Signal: x(t) Transform: Xk

〈1〉 1 δK [k]

〈2〉 ej2πTk0t δK [k − k0]

〈3〉 ∑∞m=−∞ δD(t−mT ) 1

T ∀ k

〈4〉 cos(

2πT k0t

)12(δK [k − k0] + δK [k + k0])

〈5〉 sin(

2πT k0t

)12j (δK [k − k0]− δK [k + k0])

〈6〉 ∑∞m=−∞ rect

(t−mTT0

)0 < T0 < T T0

T sinc(kT0T

)

〈7〉 ∑∞m=−∞ trian

(t−mTT0

)0 < T0 < T/2 T0

T

(sinc

(kT0T

))2

21

Discrete Fourier Transform Properties

DFT Property Periodic (N) Time Domain ←→ Periodic (N) Frequency Domain

Convolution x[n]~Ny[n] ←→ X[k]Y [k]

Linearity αx[n] + βy[n] ←→ αX[k] + βY [k]

Time Shift x[n− n0] ←→ e−j2πNkn0X[k]

Frequency Shift ej2πNk0nx[n] ←→ X[k − k0]

Inner Product∑N−1

n=0 x[n]y∗[n] = 1N

∑N−1k=0 X[k]Y [k]∗

Parseval’s Theorem∑N−1

n=0 |x[n]|2 = 1N

∑N−1k=0 |X[k]|2

Spectral Conjugation x∗[−n] ←→ X∗[k]

Autocorrelation x[n]~Nx∗[−n] ←→ |X[k]|2

Modulation x[n]y[n] ←→ 1NX[k]~NY [k]

Scaling x[rn] r ∈ Z ←→ See Section 3

Time Difference x[n]− x[n− 1] ←→(

1− e−j 2πN k)X[k]

Summation∑n

m=−∞ x[m] ←→(

1− e−j 2πN k)−1

X[k] if X[0] = 0

Real ∼ HS <{x[n]} ←→ HS {X[k]}

Imaginary ∼ HAS ={x[n]} ←→ −jHAS {X[k]}

HS ∼ Real HS {x[n]} ←→ <{X[k]}

HAS ∼ Imaginary HAS {x[n]} ←→ j={X[k]}

Duality X[n] ←→ Nx[−k]

22

Discrete Fourier Transform Pairs

No. Periodic (N) Signal: x[n] Periodic (N) Transform: X[k]

〈1〉 1 N∑∞

m=−∞ δK [k −mN ]

〈2〉 ∑∞m=−∞ δK [n−mN ] 1 ∀ k

〈3〉 ej2πNk0n N

∑∞m=−∞ δK [k − k0 −mN ]

〈4〉 ∑∞m=−∞ δK [n− n0 −mN ] ej

2πNkn0

〈5〉 cos(

2πN k0n

)N2

∑∞m=−∞(δK [k − k0 −m] + δK [k + k0 −m])

〈6〉 sin(

2πN k0n

)N2j

∑∞m=−∞(δK [k − k0 −m]− δK [k + k0 −m])

〈7〉 ∑∞m=−∞DrectN0 [n−mN ] 0 < N0 < N/2 (2N0 + 1)dincN0

(kN

)

〈8〉 ∑∞m=−∞DtrianN0 [n−mN ] 0 < N0 < N/4

((2N0 + 1)dincN0

(kN

))2

23

4 Relation Between Fourier Family Members

The development in this section is intended to complement the theoretical view of Fourier Familypresented in Section 2 by considering the real world signals which can be represented using eachtransform. The FS, DTFT and DFT are shown to be special cases of the Fourier Transform. Thesespecial cases allow us to maintain the relevant information in the signal with less overhead than theFT. An understanding of the material in this section is also required to implement a simulation ofa continuous time spectral processing system using a digital computer (or implementation in DSPhardware for that matter).

We know that any square integrable function can be represented by the FT. But this includestime-limited signals which we know can be represented using the FS. Similarly we will see that band-limited (frequency-limited) signals can be represented by the DTFT. The DFT will also be shown tobe the special case corresponding to a signal which is both time and band limited (approximately).

4.1 Band Limited Signals - DTFT and Time Sampling

If we start with a signal, x(t), which is bandlimited, that is X(f) = 0 ∀ |f | > B, then wecan completely represent the signal by a countable number of time samples. Mathematically, werepresent the time-sampled version of x(t) by

xts(t) = x(t)

∞∑

n=−∞δD(t− nTs) (86a)

xts(t) =∞∑

n=−∞x(nTs)δD(t− nTs), (86b)

where Ts = 1/fs is the sampling period. Taking the Fourier transform of both of the aboveexpressions yields two corresponding expressions for Xts(f)

Xts(f) = X(f)∗fs∞∑

n=−∞δD(f − nfs) = fs

∞∑

n=−∞X(f − nfs) (87a)

Xts(f) =∞∑

n=−∞x(nTs)e

−j2πfTsn, (87b)

where the modulation property and pair 〈4〉 have been used to get (87a) and pair 〈2〉 yields (87b).Notice that both of the expressions for Xts(f) in (87) are periodic in f with period fs. In fact (87)can be viewed as a Fourier Series in the f variable.

Since we now have a countable collection of samples, we can define a conversion to discrete timeby

24

x(t)

X(f)

xts(t)

Xts(f)

nΣ δD(t-nTs)

yn=x(nTs)

fs

Y(ν)

B-B

B-B

fs

1

fs

ν

f

f

1 B/fs-B/fs

2fsfs

-1 2

26 c⃝ K.M. Chugg - November 19, 1995

Figure 4: Conversion from continuous to discrete time (FT to DTFT)

yn = x(nTs) (88a)

Y (ν) = DTFT {yn} = X ts(f)!!!f=ν/Ts

=∞"

m=−∞fsX(fs(ν − m)). (88b)

The fact that this definition is consistent (i.e. Y (ν) really is the DTFT of yn) follows from(87b). This indicates the interpretation that you should have for the normalized frequencyν, namely that ν = fTs. This means that ν is unitless, while f is typically expressed in Hz.This conversion process is illustrated in Figure 4 for the case of fs > 2B.

The sampling theorem states that when fs ≥ 2B there is no information lost in thisconversion to discrete time. That is for fs ≥ 2B

Y (ν) = X(ν/Ts) ν ∈ [−1/2, 1/2]. (89)

Figure 4: Conversion from continuous to discrete time (FT to DTFT)

yn = x(nTs) (88a)

Y (ν) = DTFT {yn} = Xts(f)∣∣f=ν/Ts

=∞∑

m=−∞fsX(fs(ν −m)). (88b)

The fact that this definition is consistent (i.e. Y (ν) really is the DTFT of yn) follows from (87b).This indicates the interpretation that you should have for the normalized frequency ν, namely thatν = fTs. This means that ν is unitless, while f is typically expressed in Hz. This conversionprocess is illustrated in Figure 4 for the case of fs > 2B.

The sampling theorem states that when fs ≥ 2B there is no information lost in this conversionto discrete time. That is for fs ≥ 2B

Y (ν) = X(ν/Ts) ν ∈ [−1/2, 1/2]. (89)

If fs < 2B the terms in (87a) overlap and information is lost - this is what is referred to as(frequency) aliasing.

When no aliasing occurs (fs ≥ 2B) we can reconstruct the time signal from its samples. To doso, we low pass filter Xts(f)

X(f) = Xts(f)(Tsrect(fTs)). (90)

Inverting this relation (using the scaling and duality properties along with pair 〈9〉) implies

x(t) = xts(t)∗sinc(fst). (91)

25

Substituting (86b) for xts(t) and simplifying yields the interpolation formula

x(t) =∞∑

n=−∞x(nTs)sinc

(t−nTsTs

)Ts ≤ 1/2B. (92)

This interpolation can be stated in terms of the equivalent discrete signal

x(t) =∞∑

n=−∞ynsinc(2B(t− n/2B), (93)

where the minimum sampling frequency for no aliasing, the Nyquist rate, has been used (i.e.fs = 2B).

4.2 Time Limited Signals - FS and Frequency Sampling

The dual of the bandlimited time sampling problem is frequency sampling of a time limited signal.We start with a time limited signal which is centered about the origin. That is if x(t) = 0 ∀ |t| > T0,then we can completely represent the signal by a countable number of frequency samples. The stepsparallel the development of the DTFT; first we frequency sample

Xfs(f) = X(f)∞∑

k=−∞δD(f − kfs) (94a)

Xfs(f) =∞∑

k=−∞X(kfs)δD(f − kfs), (94b)

where fs = 1/Ts is the frequency sampling period in Hz. Taking the inverse FT of both of theabove expressions yields two corresponding expressions for xfs(t)

xfs(t) = x(t)∗(Ts

∞∑

n=−∞δD(t− nTs)

)= Ts

∞∑

n=−∞x(t− nTs) (95a)

xfs(t) =

∞∑

k=−∞X(kfs)e

j 2πTskn. (95b)

Thus xfs(t) is periodic in t with period Ts, so the pair in (95) can be viewed as a Fourier Series inthe t variable.

The conversion to discrete frequency is obtained through

Yk = X(kfs) (96a)

y(t) = FS−1 {y(t)} = xfs(t) =

∞∑

m=−∞Tsx(t− nTs). (96b)

This definition is consistent (i.e. Yk really is the FS of y(t)) due to (95b). This FS is withrespect to the period Ts. It is not conventional to normalize the time (i.e. τ = t/Ts), otherwise theresult would be the exact dual of (88). This conversion process is illustrated in Figure 5 for thecase of Ts > 2T0.

26

1

t

t

x(t)

X(f)Xfs(f)

Σk

δD(f-kfs)

Yk=X(kfs)

T0-T0

2Ts-Ts Ts

y(t)=xfs(t)

T0-T0

Ts

28 c⃝ K.M. Chugg - November 19, 1995

Figure 5: Conversion from continuous to discrete frequency (FT to FS)

Thus xfs(t) is periodic in t with period Ts, so the pair in (95) can be viewed as a FourierSeries in the t variable.

The conversion to discrete frequency is obtained through

Yk = X(kfs) (96a)

y(t) = FS−1 {y(t)} = xfs(t) =∞!

m=−∞Tsx(t − nTs). (96b)

This definition is consistent (i.e. Yk really is the FS of y(t)) due to (95b). This FS iswith respect to the period Ts. It is not conventional to normalize the time (i.e. τ = t/Ts),otherwise the result would be the exact dual of (88). This conversion process is illustratedin Figure 5 for the case of Ts > 2T0.

The dual of sampling theorem implies that for Ts ≥ 2T0 there is no information lost inthis conversion to discrete frequency. That is for Ts > 2T0

y(t) = Tsx(t) t ∈ [−Ts/2, Ts/2]. (97)

Figure 5: Conversion from continuous to discrete frequency (FT to FS)

The dual of sampling theorem implies that for Ts ≥ 2T0 there is no information lost in thisconversion to discrete frequency. That is for Ts > 2T0

y(t) = Tsx(t) t ∈ [−Ts/2, Ts/2]. (97)

Information is lost if Ts < 2T0 the terms in (95a) overlap - this would be time aliasing.Just as in the bandlimited case, when no aliasing occurs (Ts ≥ 2T0) the continuous variable

signal (X(f)) can be reconstructed from a countable number of samples. This is accomplished bytime windowing xfs(t)

x(t) = xfs(t)(fsrect(fst)). (98)

Inverting this relation impliesX(f) = Xfs(f)∗sinc(Tsf). (99)

Substituting (94b) for Xfs(f) and simplifying yields the frequency interpolation formula

X(f) =∞∑

k=−∞X(kfs)sinc

(f−kfsfs

)fs ≤ 1/2T0. (100)

Restating this in terms of the FS coefficients of the equivalent discrete frequency signal and samplingat the Nyquist rate (i.e. Ts = T = 2T0) yields

X(f) =

∞∑

k=−∞Yksinc(T (f − k/T )). (101)

This notation agrees with the earlier convention of y(t) being periodic with period T .

27

4.3 Approximately Time and Band Limited Signals - DFT

If we have a discrete time signal which is nonzero only for a finite number of samples, then wemight expect that we can reduce the amount of information needed to describe the spectrum. Bythe same reasoning, a similar reduction in the required information should be possible if a discretefrequency signal is fully described by a finite number of frequency samples. This is how we get tothe DFT; either by frequency sampling the DTFT of a discrete time signal or by time samplingthe inverse FS of a discrete frequency signal. We’ll use the frequency sampling approach since itseems more intuitive.

Let yn be a discrete time signal which is nonzero only for N0 consecutive values. In the previoustwo sections we centered the interval around the origin to simplify the interpolation formulas. Inthis case the interval can only be centered around the origin if N0 is odd. If N0 is odd, we willassume that yn = 0 ∀ |n| > M0, where N0 = 2M0 + 1. If N0 is even, we’ll do the next bestthing and assume that yn may be nonzero only for n ∈ {−M0,−M0 + 1, . . .M0 − 2,M0 − 1}. Thisconvention is only relevant for the interpolation formula for Y (ν).

Sampling Y (ν) will produce replicas of yn in the time domain, so that we expect to recover ynand thus Y (ν) by time windowing. The first step is to frequency sample the DTFT of yn

Y νs(ν) = Y (ν)∞∑

k=−∞δD(ν − kνs) (102a)

Y νs(ν) =∞∑

k=−∞Y (kνs)δD(ν − kνs) =

∞∑

m=−∞

Ns−1∑

k=0

Y (kνs)δD(ν − kνs −m), (102b)

where νs = 1/Ns is the normalized frequency sampling period and Ns is a positive integer. Thesecond equality in (102b) follows from the fact that any DTFT frequency function is periodic withperiod 1. Taking the inverse DTFT of both of the above expressions yields two correspondingexpressions for yνsn

yνsn = yn∗(Ns

∞∑

m=−∞δK [n−mNs]

)= Ns

∞∑

m=−∞yn−mNs (103a)

yνsn =

Ns−1∑

k=0

Y (k/Ns)ej 2πNskn, (103b)

where the convolution property and pairs 〈3〉 and 〈4〉 of the DTFT have been used. It follows thatyνsn is periodic in n with period Ns, so the pair in (103b) defines a DFT of length Ns.

Specifically, the conversion to discrete frequency is defined by

w[n] = yνsn = Ns

∞∑

m=−∞yn−mNs (104a)

W [k] = DFT {w[n]} = Y (ν)|ν=k/Ns= Y (k/Ns). (104b)

Again (103b) assures that this definition is consistent with that of the DFT.The result of normalized frequency sampling is the Ns point DFT in (104). There is no overlap

(discrete time aliasing) in (104a) if Ns ≥ N0. This process is illustrated in Figure 6. Thus the

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1

n

n

yn

Y(ν)

yn

νs

=w[n]

Y νs(ν)

Σk

δD(ν-k/Ns)

W[k]=Y(k/Ns)

Ns-Ns 2Ns-2Ns

Ns

M0-M0

M0-M0

Transform Theory Notes 31

Figure 6: Conversion from discrete time frequency to discrete frequency (DTFT to DFT)

This simplifies when we take the minimum sampling rate (Ns = N0 = N) and use theconversion in (104b)

Y (ν) =N−1!

k=0

W [k]dincM (ν − k/N) N = 2M + 1. (108)

A similar argument can be used when Ns is even. Using the convention that yn is nonzero only for n ∈ {−M, −M + 1, . . . M − 1} and sampling at the minimum rate yields

Y (ν) =N−1!

k=0

W [k]

"dincM (ν − k/N) − (−1)k

Ne−jπNν

#N = 2M. (109)

The phase factor arises from the fact that the interval is not centered about the origin.9

9In most references there is always a complex phase factor in the interpolation formula because they usethe interval {0, 1, . . . N − 1}.

Figure 6: Conversion from discrete time frequency to discrete frequency (DTFT to DFT)

spectrum of a discrete time signal with at most N0 (consecutive) nonzero samples can be representedby as few as N0 samples of the DTFT. To get an interpolation formula for Y (ν) in terms of thesamples (with Ns ≥ N0), we time window. Consider the case when Ns is odd (i.e. Ns = 2Ms + 1)

yn = yνsn (DrectMs [n]/Ns) . (105)

Taking the DTFT, using the modulation property and pair 〈9〉, yields

Y (ν) = Y νs(ν)~1dincMs (ν), (106)

which simplifies, using (102b), to

Y (ν) =

Ns−1∑

k=0

Y (k/Ns)dincMs (ν − k/Ns) Ns = 2Ms + 1 ≥ N0. (107)

This simplifies when we take the minimum sampling rate (Ns = N0 = N) and use the conversionin (104b)

Y (ν) =

N−1∑

k=0

W [k]dincM (ν − k/N) N = 2M + 1. (108)

A similar argument can be used when Ns is even. Using the convention that yn is non zero onlyfor n ∈ {−M,−M + 1, . . .M − 1} and sampling at the minimum rate yields

Y (ν) =

N−1∑

k=0

W [k]

(dincM (ν − k/N)− (−1)k

Ne−jπNν

)N = 2M. (109)

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The phase factor arises from the fact that the interval is not centered about the origin.9

4.4 The Time and Bandlimited Paradox

It initially appears that the development of the previous three sections is a recipe to represent a timelimited and bandlimited signal by a finite collection of numbers. If a signal, x(t), is bandlimited wecan time sample and represent it by a countable number of samples (i.e. convert to the discrete timesignal yn). If this discrete time signal is time limited we can frequency sample and represent thespectrum with a finite number of points. If the original signal had bandwidth B(i.e. f ∈ [−B,B])and duration T (i.e. t ∈ [−T/2, T/2]), then it would be possible to completely represent the signalby 2BT samples.

The catch is that no signal is strictly bandlimited and time limited! If the signal is time limited,then it can be time windowed without affecting the signal. This time windowing corresponds toconvolution with a sinc function in the frequency domain, which implies that the signal spectrum isspread over the entire frequency axis. The reverse argument can be used to see that a bandlimitedsignal is not time limited. Thus whenever one applies a DFT to approximate the FT of a real worldsignal there is an approximation being made. In fact we can combine the conversions made in (88)and (104) to get the DFT pair in terms of the original FT pair,10

w[n] = Ns

∞∑

m=−∞x((n−mNs)Ts) (110a)

W [k] = DFT {w[n]} =1

Ts

∞∑

m=−∞X

(k −mNs

NsTs

). (110b)

Since x(t) cannot be both time and freqeuncy limited, there must be overlap (aliasing) in atleast one of the summations in (110). Obviously, the DFT has found great use in modern signalprocessing systems, so the approximation error can be made acceptably small for most applicationsby sampling at high rates.

The results of this section are summarized in Figure 7. The diagram in Figure 7 also includes theZ, Laplace and Fast Fourier Transform (FFT). The Z and Laplace transforms are briefly discussedin the next section. The FFT is just a computational algorithm for computing the DFT (sometimesthe terms “DFT” and “FFT” are used interchangably).

5 Laplace and Z Transforms

The (two sided) Laplace and Z transforms can be viewed as extensions of the FT and DTFTrespectively. We are concerned with these transforms primarily as tools to recognize a stable causalsystem in the frequency domain.

9In most references there is always a complex phase factor in the interpolation formula because they use theinterval {0, 1, . . . N − 1}.

10This is not the only way to approximate a real signal by a DFT, in fact it is not even a practical approach sinceit requires values of x(t) for all time. The approach taken most often in practice is to time sample, then time windowand perform a DFT.

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Fourier Transform• Continuous Time• Continuous Frequency

Fourier Series • Continuous Time (periodic)• Discrete Frequency

Discrete Fourier Transform (DFT)• Discrete Time (periodic)• Discrete Frequency (periodic)

Discrete Time Fourier Transform (DTFT)• Discrete Time• Continuous Frequency (periodic)

Duality

Time-Limited Signal Frequency

SamplingTime

Sampling

Band-Limited Signal

Discrete Band-Limited

Signal

Time Sampling

Frequency Sampling

Discrete Time-Limited

Signal

Duality

Duality

Laplace Transform

Z Transform

Fast Fourier Transform (FFT)

s = j2⇡f

z = esTs

z = ej2⇡⌫

Computational Algorithm

Figure 7: Relation between various transforms.

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5.1 Z Transform of a Stable, Causal System

The Z transform of a discrete time signal is defined as

H(z) =

∞∑

n=−∞hnz

−n z ∈ ROC (111a)

hn =1

2πj

∮

Czn−1H(z)dz C closed, counterclockwise in ROC. (111b)

A Z transform is incomplete without the region of convergence (ROC). The ROC of a Z transformis always of the form

ROC = {z : a < |z| < b}. (112)

If the sequence is absolutely summable (corresponding to a stable system impulse response) theDTFT also exists, which implies that the unit circle is in the ROC since11

H(ν) = H(z)|z=exp(j2πν) . (113)

If the sequence is right-sided (i.e. hn = 0 ∀ n < n0) then b = ∞. For a causal system “thepoint” z =∞ is also in the ROC, since for a causal system

H(z) = h0 + h1z−1 + h2z

−2 + h3z−3 . . . (114)

which converges as z →∞. This yields the initial value theorem for causal systems

h0 = limz→∞

H(z). causal systems (115)

5.2 Laplace Transform of a Stable, Causal System

The Laplace transform of a continuous time signal is defined as

H(s) =

∫ ∞

−∞h(t)e−stdt s ∈ ROC (116a)

h(t) =1

2πj

∫ σ+j2π∞

σ−j2π∞H(s)estds σ + j2πf ∈ ROC (116b)

A Laplace transform is also incomplete without the ROC. The ROC of a Laplace transform isalways of the form

ROC = {s : a < <{s} < b}. (117)

If the signal is absolutely integrable, corresponding to a stable system impulse response, the FTalso exists, which implies that the imaginary axis is in the ROC since12

H(f) = H(s)|s=j2πf . (118)

11This notation should be not be interpreted as implying that H(ν), the DTFT, is the Z transorm evaluated atz = ν. Rather it is the Z transform evaluated on the unit circle.

12Again this notation should not be interpreted literally. The FT is the Laplace transform evaluated on theimaginary axis (i.e. s = j2πf); it is not obtained by substituting f for s.

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There is also an initial value theorem for causal continous time systems

h(0+) = lims→∞

sH(s), causal systems (119)

where h(0+) is the limit as t ↓ 0.

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