fourier supplementals

1

Supplementary Note for Signals and Systems:

The Four Fourier Representations

Shang-Ho (Lawrence) Tsai

Department of Electrical Engineering

College of Electrical and Computer Engineering

National Chiao Tung University

Hsinchu Taiwan

E-mail: [email protected]

CONTENTS

I Goal of this supplemental note 2

II Represent a signal by combining weighted sinusoids 2

II-A When x is periodic and discrete . . . . . . . . . . . . . . . . . . . . . . . 4

II-B When x is periodic and continuous . . . . . . . . . . . . . . . . . . . . . . 5

II-C When x is nonperiodic and discrete . . . . . . . . . . . . . . . . . . . . . 5

II-D When x is nonperiodic and continuous . . . . . . . . . . . . . . . . . . . . 6

III Determine the weights (coefficients) of sinusoids 7

III-A When x is periodic and discrete . . . . . . . . . . . . . . . . . . . . . . . 9

III-B When x is nonperiodic and discrete . . . . . . . . . . . . . . . . . . . . . 10

III-C When x is periodic and continuous . . . . . . . . . . . . . . . . . . . . . . 11

III-D When x is nonperiodic and continuous . . . . . . . . . . . . . . . . . . . . 11

References 12

April 13, 2009 DRAFT

2

I. GOAL OF THIS SUPPLEMENTAL NOTE

This supplemental note is to help students who take “Signals and Systems” to clarify the

fundamental concept in Fourier representations. Moreover, it tries to help students to build an

intuition why the Fourier and the inverse representations are defined like that in the textbook.

After reading this note, students are expected to be able to answer the following questions:

• Why are the Fourier representations for discrete signals always periodic (DTFS and DTFT)?

• Why are the Fourier representations for periodic signals always discrete (FS and DTFS)?

• Why are the integration intervals for the Fourier representations sometimes from −∞ to ∞

and sometimes from −π to π (FT and DTFT)?

• Why are the integration intervals for the inverse Fourier representations sometimes from

−∞ to ∞ and sometimes from 0 to T (IFT and IFS)?

• Why are the summation elements for the Fourier representation sometimes from −∞ to ∞

and sometimes from 0 to N − 1 (FS and DTFS)?

• Why are the summation elements for the inverse Fourier representations sometimes from

−∞ to ∞ and sometimes from 0 to N − 1 (IDTFT and IDTFS)?

• Why are the normalization factors sometimes N , sometimes 2π, sometimes T , and some-

times 1 (For all four Fourier and inverse Fourier representations)?

• Why are the units for frequencies sometimes Hz, sometimes rad/sec and sometimes rad?

II. REPRESENT A SIGNAL BY COMBINING WEIGHTED SINUSOIDS

Since the existing problems of the Fourier representations are beyond our scope, we simply

assume the Fourier representations exist for the signals that we discuss herein. The basic concept

of the Fourier representation is to represent a signal x by combining (sum or integral) several

weighted sinusoids which are discrete or continuous in frequency.

Question 1. Suppose that x is periodic, are the frequencies of the sinusoids to represent x

discrete or continuous?

Solution: Let us consider that x is a discrete signal with fundamental period N first. In this

case, the sinusoids are discrete in time and are periodic. Let the periods of the sinusoids be Nk.

Since x is periodic, to represent x, all the periods of the sinusoids must satisfy

N = kNk, (1)


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where k must be an integer equal or greater than 1. If (1) is not satisfied, the sinusoids do not

have a common period of N . As a result, there is no way that x can be completely represented

by the sinusoids. From (1), the relationship of the frequencies between x and the sinusoids is

given by

1/N = 1/(kNk). (2)

Define the fundamental (angle) frequency of x as

Ω0 = 2π/N (rads), (3)

and the (angle) frequencies of the sinusoids as

Ωk = 2π/Nk (rads). (4)

From (2), (3) and (4), we have

kΩ0 = Ωk. (5)

From (5) and using the fact that k is an integer, we know that the (angle) frequencies of the

sinusoids are discrete.

Next, let us consider that x is a continuous signal with fundamental period T . In this case,

the sinusoids are also continuous in time and are periodic. By using similar argument as that

for discrete signal, we know that the periods of the sinusoids must also be multiples of T , i.e.

T = kTk, (6)

where Tk are the periods of the sinusoids and k must again be an integer equal or greater than

1. From (6), the relationship of frequencies between x and the sinusoids is given by

1/T = 1/(kTk) (Hz). (7)

Define the fundamental (angle) frequency of x as

ω0 = 2π/T (rads/sec), (8)

and the (angle) frequencies of the sinusoids as

ωk = 2π/Tk (rads/sec). (9)

From (7), (8) and (9), we have

kω0 = ωk. (10)


4

From (10) and using the fact that k is an integer, we know that the (angle) frequencies of the

sinusoids are discrete. From the discussion above, we can conclude the following property:

Property 1: The frequencies of the sinusoids to represent a period signal are discrete. We

usually say that a periodic signal have discrete frequency spectrum.

From Property 1, we know that to represent a periodic signal, the frequencies of the sinusoids

are discrete. Now we can write down the relationship between x and the sinusoids. Let us discuss

the relationship for discrete x and continuous x separately as follows:

A. When x is periodic and discrete

In this case, let us rewrite x as x[n] to reflect the fact that x is discrete in time. Since x[n]

is discrete in time, the sinusoids to represent x[n] should also be discrete in time. Since x[n] is

periodic, from Property 1 and (5), we let the sinusoids be ejkΩ0n. Then, x[n] can be represented

by the combination of weighted sinusoids ejkΩ0n, we have

x[n] =∑

k

X[k]ejkΩ0n, (11)

where X[k] are the weights (or coefficients) at frequency kΩ0. To determine the range of k in

(11), we are asked how many different frequencies are sufficient to represent x[n]. To answer

this question, we notice that Ω0 = 2π/N . Hence, we have

ejkΩ0n = ej 2πN

kn. (12)

From (12), we observe that ejkΩ0n is periodic for k with period N , i.e.

ej(k+N)Ω0n = ej 2πN

(k+N)n = ej 2πN

knej 2πN

Nn = ej 2πN

kn = ejkΩ0n. (13)

Hence, from (13), we know that given a fundamental frequency Ω0 = 2π/N , we can generate

N different frequencies of sinusoids for all integer k, i.e.

ej0Ω0n, ej1Ω0n, ej2Ω0n, · · · , ej(N−1)Ω0n. (14)

Thus, it is sufficient to use N sinusoids in (17) to represent x[n]. Therefore, we can rewrite (11)

as

x[n] =N−1∑

k=0

X[k]ejkΩ0n, (15)

which leads to the definition of inverse Discrete-time Fourier Series (IDTFS).


5

B. When x is periodic and continuous

In this case, let us rewrite x as x(t) to reflect the fact that x is continuous in time. Since x(t)

is continuous in time, the sinusoids to represent x(t) should also be continuous in time. Since

x(t) is periodic, from Property 1 and (10), we let the sinusoids be ejkω0t. Then, x(t) can be

represented by the combination of weighted sinusoids ejkω0t, we have

x(t) =∑

k

X[k]ejkω0t, (16)

where X[k] are the weights (or coefficients) at frequency kω0. Now let us determine the range

of k in (16). From (16), the values ejkω0t = ejk 2πT

t for different k are different, since there is no

particular relationship between k and T . Hence, we do not have similar results as that in IDTFS

from Eqs. (11) to (15). Therefore, we know that given a fundamental frequency ω0 = 2π/T , we

can generate infinite different frequencies of sinusoids for all integer k, i.e.

· · · , e−j2ω0n, e−j1ω0n, ej0ω0n, ej1ω0n, ej2ω0n, · · · . (17)

Thus, we need infinite frequencies of sinusoids to represent x(t). Therefore, we can rewrite (16)

as

x(t) =∞∑

k=−∞

X[k]ejkω0t, (18)

which is the definition of inverse Fourier Series (IFS).

What happens if x is nonperiodic? Under such situation, Property 1 no longer holds. Hence,

the frequencies of the sinusoids are no longer multiples of the fundamental frequencies. Thus,

the frequencies of the sinusoids are continuous if x is nonperiodic. We usually say that the

frequency spectrum of a nonperiodic signal is continuous. In this case, to represent x, we need

to use integration instead of summation to combine the sinusoids. Again, x can be discrete or

continuous. Let us discuss these two cases separately as follows:

C. When x is nonperiodic and discrete

In this case, let us rewrite x as x[n] to reflect the fact that x is discrete in time. Since x[n]

is discrete in time, the sinusoids to represent x[n] should also be discrete in time. In addition,

now x[n] is nonperiodic, hence the frequencies of the sinusoids are continuous. From discussion


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above, it is reasonable to let the sinusoids be ejΩn, where the unit of Ω is rads. Thus, x[n] can

be represented by the combination of weighted sinusoids given by

x[n] =

∫

Ω

A(Ω)ejΩndΩ, (19)

where A(Ω) is the weight for sinusoid at frequency Ω. To determine the range of Ω in (19), we

notice that since n is an integer, we know ejΩn is periodic for Ω with period 2π, i.e.

ej(Ω+2π)n = ejΩnej2πn = ejΩn.

Hence, it is sufficient to use the sinusoids with frequencies in the range where −π < Ω ≤ π to

represent x[n]. Hence, we can rewrite (19) as

x[n] =

∫ π

−π

A(Ω)ejΩndΩ. (20)

By defining new weights as

X(ejΩ) = 2πA(Ω), (21)

we can rewrite (20) as

x[n] =1

2π

∫ π

−π

X(ejΩ)ejΩndΩ, (22)

which is the definition of inverse Discrete-time Fourier Transform (IDTFT). Please note that the

reason to write the new weights as X(ejΩ) instead of X(Ω) is to reflect the fact that X(ejΩ)

is periodic with period 2π. We will explain later why X(ejΩ) is period. In addition, we will

explain later why we need to define the new weights.

D. When x is nonperiodic and continuous

In this case, let us rewrite x as x(t) to reflect the fact that x is continuous in time. Since

x(t) is continuous in time, the sinusoids to represent x(t) should also be continuous in time. In

addition, now x[n] is nonperiodic, hence the frequencies of the sinusoids are continuous. From

discussion above, it is reasonable to let the sinusoids be ejωt, where the unit of ω is rad/sec.

Thus, x(t) can be represented by the combination of weighted sinusoids given by

x(t) =

∫

ω

A(ω)ejωtdω, (23)


7

where A(ω) is the weight for sinusoid at frequency ω. Please note since t is not an integer,

different values for ω will lead to different frequencies of sinusoids ejωt. Hence, we need to use

the sinusoids with all possible ω to represent x(t). Hence, we rewrite (23) as

x(t) =

∫ ∞

−∞

A(ω)ejωtdω, (24)

By defining new weights as

X(jω) = 2πA(ω), (25)

we can rewrite (24) as

x(t) =1

2π

∫ ∞

−∞

X(jω)ejωtdω, (26)

which is the definition of inverse Fourier Transform (IFT). We will explain later why we need

to define the new weights.

III. DETERMINE THE WEIGHTS (COEFFICIENTS) OF SINUSOIDS

In the previous section, we already introduced four different Fourier representations to repre-

sent signals by combining weighted sinusoids. Except the weights, we actually have everything

to represent signals using sinusoids. In this section, we would like to determine the weights. The

detailed derivation of how to determine the weights is beyond our scope. Instead, we give simple

concept to help students gain insight into how to determine the weights. First, let us introduce

the concept of basis.

Definition: A basis B of a vector space V is a linearly independent subset of V that spans (or

generates) V .

Theorem: If B (b1, b2, · · · ) is an orthogonal basis (B is a basis and the vectors in B are

orthogonal) and span the vector space V , the unique representation of any vector v in V can be

expressed as

v =∑

k

b†kv

︸︷︷︸

ck

bk, (27)

where b†k is complex-conjugate of bk.


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From (27), we can regard ck = b†kv as the projection of v on bk. Since v is a linear

combination of bk, we can also regard this projection ck as a “weight” of bk to represent

v. In this case, we may explain (27) as that vector v is the sum of weighted bk.

Example: The most famous orthogonal basis used in our real life is the Cartesian coordinate

system. The basis B in the three-dimension Cartesian coordinate system is define as

b1 = (1 0 0)t, b2 = (0 1 0)t, and b3 = (0 0 1)t.

According to the theorem, a vector v = (2 3 4)t can be written as

v = b†1vb1 + b

†2vb2 + b

†3vb3 = 2b1 + 3b2 + 4b3,

where 2 is the weight for b1, 3 is the weight for b2 and 4 is the weight for b3, to represent v.

Similar to the vector space, for continuous functions, we may regard the basis B as that it

contains several vectors (may be finite or infinite) and each vector has infinite elements. Since the

functions are continuous, we can use similar representation in (27) and represent a continuous

function v(t) as

v(t) =∑

k

∫

τ

b∗k(τ)v(τ)dτ

︸︷︷︸

ck

bk(t), (28)

where b∗k(τ) is conjugate of bk(τ). From (28), we can regard ck =∫

τb∗k(τ)v(τ)dτ as the projection

of v(t) on bk(t). Since v(t) is obtained by combining bk(t), we can also regard this projection

ck as a “weight” of bk(t) to represent v(t). In this case, we may explain (28) as that function

v(t) is the sum of weighted bk(t).

When the signal x(t) or x[n] is nonperiodic, the weight is no longer discrete. In this case, we

will replace ck by cf , where f is a continuous function of f . Hence, the expression in (28) is

rewritten as

v(t) =

∫

f

∫

τ

b∗f (τ)v(τ)dτ

︸︷︷︸

cf

bf (t)df, (29)

With the concept of (27), (28) and (29), we can explain how to obtain the weights for the

four Fourier representations.


9

A. When x is periodic and discrete

Referring to (15), our goal is to determine the weights X[k]. Comparing (15) and (27), we

can regard ejkΩ0n as an orthogonal basis to represent x[n]. Hence, referring to (27), the weights

of ejkΩ0n are obtained by projecting x[n] on ejkΩ0n, i.e.

ck =∑

n

x[n]e−jkΩ0n. (30)

Since x[n] is periodic with period N , the sinusoids ejkΩ0n also have a common period of N , the

projection only involves a period of N elements for x[n] and ejkΩ0n. We can rewrite (30) as

ck =N−1∑

n=0

x[n]e−jkΩ0n. (31)

To make the pair of DTFS and IDTFS lead to an identity operator, i.e.

x[n] = IDTFS DTFS x[n] ,

we define new weights according to (31) as

X[k] =1

N

N−1∑

n=0

x[n]e−jkΩ0n, (32)

which leads to the weight definition of DTFS, i.e DTFS. It can be easily verified that the IDTFS

and DTFS pair in (15) and (32) indeed leads to an identity operator.

Question 2. Is X[k] in (32) periodic or nonperiodic?

Solution: Since n is an integer, e−jkΩ0n = e−j 2πN

kn is periodic for k with period N , i.e.

e−j(k+N)Ω0n = e−j 2πN

(k+N)n = e−j 2πN

kne−j 2πN

Nn = e−j 2πN

kn = e−jkΩ0n,

we know that X[k] in (32) is also periodic with period N , i.e.

X[k + N ] = X[k]. (33)


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B. When x is nonperiodic and discrete

Referring to (22), our goal is to obtain the weight X(ejΩ). Comparing (22) and (27), we can

regard ejΩn as a basis to represent x[n]. Hence, referring to (27), the weights of ejΩn are obtained

by projecting x[n] on ejΩn, i.e.

c(Ω) =∑

n

x[n]e−jΩn. (34)

Note that since the frequency spectrum of a nonperiodic signal is continuous, it is more appro-

priate to use c(Ω) instead of ck to represent the weights in (34). Since x[n] is nonperiodic, the

projection in (34) involves infinite elements for x[n] and ejΩn. Hence we can rewrite (34) as

c(Ω) =∞∑

n=−∞

x[n]e−jΩn. (35)

To make the pair of DTFT and IDTFT lead to an identity operator, i.e.

x[n] = IDTFT DTFT x[n] ,

we already multiplied 1/2π in the definition of IDTFT in (22). Hence, there is no need to add

normalization factor for the weights in (35). It can be easily verified that the IDTFT and DTFT

pair in (22) and (35) indeed leads to an identity operator.

Question 3. Is c(Ω) in (35) periodic or nonperiodic?

Solution: Since n is an integer, e−jΩn = e−j(Ω+2π)n is periodic for Ω with period 2π, i.e.

e−j(Ω+2π)n = e−jΩne−j2πn = e−jΩn,

we know that c(Ω) is periodic with period 2π, i.e.

c(Ω + 2π) = c(Ω). (36)

To reflect the fact that the weights of DTFT are periodic, we define a new weight X(ejΩ) =

c(Ω). Then, we have

X(ejΩ) =∞∑

n=−∞

x[n]e−jΩn, (37)

which leads to the weight definition of DTFT, i.e DTFT.

From Question 2 and Question 3, we can conclude the following important property.

Property 2: The weights of Fourier representations for discrete signals are periodic. We

usually say that the frequency spectrum of a discrete signal is periodic.


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C. When x is periodic and continuous

Referring to (18), our goal is to obtain the weights X[k]. Comparing (18) and (28), we can

regard ejkω0t as an orthogonal basis to represent x(t). Hence, referring to (28), the weights of

ejkω0t are obtained by projecting x(t) on ejkω0t, i.e.

ck =

∫

t

x(t)e−jkω0tdt. (38)

Since x(t) is periodic with period T and the sinusoids ejkω0t also have a common period of T ,

the projection only involves a period of T for x(t) and ejkω0t. Hence we can rewrite (38) as

ck =

∫ T

0

x(t)e−jkω0tdt. (39)

To make the pair of FS and IFS lead to an identity operator, i.e.

x(t) = IFS FS x(t) ,

we define new weights according to (39) as

X[k] =1

T

∫ T

0

x(t)e−jkω0tdt, (40)

which leads to the weight definition of FS, i.e FS. It can be easily verified that the IFS and FS


D. When x is nonperiodic and continuous

Referring to (26), our goal is to obtain the weight X(jω). Comparing (26) and (29), we can

regard ejωt as a basis to represent x(t). Hence, referring to (29), the weights of ejωt are obtained

by projecting x(t) on ejωt, i.e.

c(ω) =

∫

t

x(t)e−jωtdt. (41)

Note that since the frequency spectrum of a nonperiodic signal is continuous, it is more appro-

priate to use c(ω) instead of ck to represent the weights in (41). Since x(t) is nonperiodic, the

projection in (41) involves infinite period for x(t) and ejωt. Hence we can rewrite (41) as

c(ω) =

∫ ∞

−∞

x(t)e−jωtdt. (42)

To make the pair of FT and IFT lead to an identity operator, i.e.

x(t) = IFT FT x(t) ,


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we already multiplied 1/2π in the definition of IFT in (26). Hence, there is no need to add

normalization factor for the weights in (42). Letting X(jω) = c(ω), we have

X(jω) =

∫ ∞

−∞

x(t)e−jωtdt, (43)

which leads to the weight definition of FT, i.e FT. It can be easily verified that the IFT and FT


REFERENCES

[1] S. Haykin and B. V. Veen, Signals and Systems, 2nd Edition, John Wiley & Sons, Inc., 2003.

[2] G. Strang, Linear Algebra and Its Applications, 3nd Edition, Brooks Cole, Inc., 1988.
