Spectrum Representation (5A) - Wikimedia€¦ · 28/10/2016 · 5A Spectrum Representation 5 Young Won Lim 10/28/16 Fourier Series with complex coefficients x(t) = ∑ k=−∞ C

Young Won Lim10/28/16

Spectrum Representation (5A)


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5A Spectrum Representation

3 Young Won Lim10/28/16

Fourier Series with real coefficients

x (t ) = a0 + ∑k=1

∞

(ak cos(kω0 t) + bk sin(kω0 t ))

x(t ) = g0 + ∑k=1

∞

gk cos(kω0 t + ϕk)

x (t ) = X 0 + ∑k=1

∞

ℜ{X k e+ j kω0 t}

1

a

b

X k = gk⋅e+ jϕk



Phasor Representation

x(t ) = g0 + ∑k=1

∞


x (t ) = g0 + ∑k=1

∞

gk ℜ{e+ j (kω0 t + ϕk)}

x (t ) = g0 + ∑k=1

∞

ℜ{gk⋅e+ jϕk⋅e+ j kω0 t}

X k via gk , ϕk

X k = gk⋅e+ jϕk



Fourier Series with complex coefficients

x (t ) = ∑k=−∞

+∞

C k e+ j kω0 t

x(t ) = g0 + ∑k=1

∞


x (t ) = X 0 + ∑k=1

∞

ℜ{X k e+ j kω0 t}

4

a

b

Ck =12g+k e

+ jϕk (k > 0) C k =12g−k e− jϕk (k < 0)



Complex Coefficient

x (t ) = ∑k=−∞

+∞

C k e+ j kω0 t

4

Ck =12g+k e

+ jϕk (k > 0) C k =12g−k e− jϕk (k < 0)

= g0 + ∑k=1

∞

gk⋅12

(e+ j(kω0 t + ϕk) + e− j(kω0 t + ϕk))

= g0 + ∑k=1

∞

([ 12 gk e+ jϕk ] e+ j k ω0 t + [ 12 gk e− jϕk ]e− j kω0 t)= g0 + ∑

k=1

∞

( [C k ] e+ j kω0 t + [C−k ]e

− j kω0 t )

Ck via gk , ϕk

x (t) = g0 + ∑k=1

∞

gk⋅cos(kω0 t + ϕk)

⋯ , C−1 , C0 , C+1 , ⋯

g0 , g1 , g2 , ⋯



Fourier Coefficients Relationship

a0 =1T ∫0

Tx (t) dt

ak =2T ∫0

Tx (t) cos (kω0 t) dt

bk =2T ∫0

Tx (t) sin (kω0 t) dt

g0 = a0

gk = √ak2 + bk

2

ϕk = tan−1 (−bk

ak)

X 0 = g0

X k = gk⋅e+ jϕk

1 a

b

k = 1, 2, .. .

Ck = 12 g+k e

+ jϕk (k > 0)

C k = 12 g−k e

− jϕk (k < 0)

4

k = 1, 2, .. .

k = 1, 2, .. .k = 0, ±1, ±2, .. .



Single-Sided Spectrum

x(t) = a0 + ∑k=1

∞

(ak cos(k ω0 t) + bk sin(k ω0 t)) x(t) = g0 + ∑k=1

∞

g k cos(k ω0 t + ϕk )

a0 =1T ∫0

Tx( t) dt

a k =2T∫0

Tx (t) cos (k ω0 t) dt

bk =2T∫0

Tx (t ) sin (k ω0 t) dt

g0 = a0

gk = √ ak2+ bk

2


ak)

a0 , a1 , a2 , ...b0 , b1 , b2 , ...

g0 , g1 , g2 , ...ϕ0 , ϕ1 , ϕ2 , ...



Single-Sided Spectrum

x(t) = a0 + ∑k=1

∞

(ak cos(k ω0 t) + bk sin(k ω0 t)) x(t) = g0 + ∑k=1

∞

g k cos(k ω0 t + ϕk )

= cos (α) cos(β)

gk cos(k ω0 t + ϕk ) = gk cos (ϕk ) cos(k ω0 t) − gk sin (ϕk) sin (k ω0 t)

cos(α+β) − sin (α) sin(β)

ak cos(kω0 t ) + bk sin(kω0 t )

a k = g k cos(ϕk)

−bk = g k sin (ϕk )

g0 = a0

gk = √ ak2+ bk

2


ak)

a0 = g0

a k2+ bk

2= g k

2

−bk

ak

= tan (ϕk)



Periodogram

x(t ) = g0 + ∑k=1

∞


a

Periodogram

|gk| = √ak2 + bk

2

One-Sided

g0 , g1 , g2 , ⋯



Power Spectrum

x (t ) = ∑k=−∞

+∞

C k e+ j kω0 t

4

Power Spectrum

|C k|2 = 1

4 (ak2 + bk

2)

Two-Sided

⋯, C−1 , C0 , C+1 , ⋯



Power Spectrum

x (t ) = ∑k=−∞

+∞

C k e+ j kω0 t

4

Power Spectrum

|C k|2 = 1

4 (ak2 + bk

2)

P =1T∫T

|x (t )|2dt

=1T∫T

x (t)x∗(t )dt

=1T∫T

x∗(t) ∑

k=−∞

+∞

Ck e+ j kω0 t dt

=1T

∑k=−∞

+∞

C k∫T

x∗(t) e+ j kω0 t dtC k =

1T∫0

T

x (t) e− j kω0 t dt

C k∗

=1T∫0

T

x∗(t) e

+ j kω0 t dt =1T

∑k=−∞

+∞

C kT C k∗

=|C k|2



Two-Sided Spectrum

x ( t) = g0 + ∑k=1

∞

gk cos(kω0 t + ϕk)x (t ) = a0 + ∑k=1

∞

(ak cos(k ω0 t ) + bk sin(kω0 t ))

x ( t) = ∑k=−∞

+∞

C k e+ j kω0 t

Ck =

a0 (k = 0)12 (ak − j bk) (k > 0)12 (ak + j bk) (k < 0)

|C k| =a0 (k = 0)12 √ak

2 + bk2 (k ≠ 0)

Arg(C k ) =tan−1 (−bk /ak ) (k > 0)

tan−1 (+bk /ak ) (k < 0)

x (t) = ∑k=−∞

+∞

C k e+ j kω0 t

Ck =

g0 (k = 0)12 g+k e

+ jϕk (k > 0)12 g−k e− j ϕk (k < 0)

|C k| =g0 (k = 0)12|gk| (k ≠ 0)

Arg(C k ) = +ϕk (k > 0)

−ϕk (k < 0)

Power Spectrum Periodogram

|C k|2 = |C−k|

2 = 14|gk|

2 = 14 (ak

2 + bk2) |gk| = 2⋅|C k| = √ak

2 + bk2

Two-Sided One-Sided



Spectrum

Real Single-tone Sinusoidal Signal

x(t) = A cos(ω0 t + ϕ)

= ℜ{X e jω0 t}

X = A e jϕ

=A2

(e+ j(ω0 t + ϕ)+ e− j(ω0 t + ϕ)

)

=A2

(e+ jϕe+ jω0 t + e− jϕe− jω0 t)

={A e+ jϕ

}

2e+ jω0 t +

{A e+ jϕ}

2

∗

e− jω0 t= {X2 e+ jω0t +X∗

2e− jω0t }

A cos(ω0 t + ϕ)



Spectrum

x(t) = A0 + ∑k=1

N

A k cos(ωk t + ϕk)

Real Multi-tone Sinusoidal Signal

= X 0 + ℜ{∑k=1

N

X k ejωk t }

X 0 = A0

= X 0 + ∑k=1

N

{X k

2e+ jωk t +

X k∗

2e− jωk t}

the phasor of angular frequency ωk

X k = A k ejϕk

Real Single-tone Sinusoidal Signal

x(t) = A cos(ω0 t + ϕ)

= ℜ{X e jω0 t}

X = A e jϕ

= {X2 e+ jω0t +X∗

2e− jω0t }



Spectrum

only magnitude is display

X k

2

X k∗

2

X k

2e+ jωk t

X k∗

2e− jωk t

−ωk +ωk

x(t) = A0 + ∑k=1

N

A k cos(ωk t + ϕk)

Real Multi-tone Sinusoidal Signal

= X 0 + ℜ{∑k=1

N

X k ejωk t }

X 0 = A0

= X 0 + ∑k=1

N

{X k

2e+ jωk t +

X k∗

2e− jωk t}

the phasor of angular frequency ωk

X k = A k ejϕk



Frequency Resolution of Fourier Series Representations

+∞−∞

+π−π

ω̂0 ω̂0

+∞0

0

0

0

T s T s

CTFS

DTFS / DFT

ω0 ω0

ω0 =2π

T 0

ω̂0 =2π

N 0

period : T 0 seconds

period : N0 samples

Continuous Time

Discrete Time

Discrete Frequency

Normalized Discrete Frequency

Fourier Analysis Overview (0B) 18 Young Won Lim

10/28/16

Sampling Period and the Number of Samples

x (t )

T 0 period

N 0 samplesT s

x [n]

ω0 =2π

N0T s

ω0 =2π

T 0

ω0

1/T s

=2πN 0T s

T s

ω̂0 =2π

N 0

normalized frequency resolution

frequency resolution

T 0 = N 0T s

⋅T s ⋅ f s

Fourier Analysis Overview (0B) 19 Young Won Lim

10/28/16

Sampling Period and the Number of Samples

x (t )

T 0 period

N 1 samples

N 2 samples

N 1 < N2

T 1 > T 2

x1[n]

x2[n]

T 1N1 = T 2N2

ωs =2π

T s

2πT 1

2πT2

ω1

ω2

ωs = ω1

ωs = ω2

ω0T 1 > ω0T 2

T s = T 1

T s = T 2

ω1 < ω2

ω0 =ω̂0

T s



Frequency and Digital Frequency

ω̂ (rad /sample)

ω (rad /sec )

Frequency

Digital Frequency

−ωs +ωs−ω0 +ω0−ω0−ωs +ω0+ωs

−ω̂0 +ω̂0 +ω̂0+2π−ω̂0−2π +2π−2π

e− jω s t e+ jωs te− jω 0 t e+ jω0 te− j (ω0+ω s)t e+ j(ω0+ωs) t

e− j ω̂ 0n e+ j ω̂0 n e+ j( ω̂0+2π )ne− j (ω̂0−2π )n e+ j+2πe− j2π n



Frequency and Digital Frequency

ω̂ = ω⋅T s

ω̂ = ωf s

tT s

T 0

x(t )= cos(ω0 t) x [n] = x(nT s)= cos(nω0T s)



γ[k ] =1N

∑n = 0

N−1

x [n] e− j k ω̂0n x [n] = ∑k = 0

N−1

γ[k ] e+ j k ω̂0n

Fourier Transform Types

Continuous Time Fourier Series

C k =1T∫0

Tx (t ) e− j k ω0 t dt

Discrete Time Fourier Series

x (t ) =12π

∫−∞

+∞

X ( jω) e+ jω t dω

Continuous Time Fourier Transform

x [n] =12π

∫−π

+π

X ( j ω̂) e+ j ω̂nd ω̂

Discrete Time Fourier Transform

X ( j ω̂) = ∑n =−∞

+∞

x [n] e− j ω̂n

x (t ) = ∑k=−∞

+∞

Ck e+ j kω0 t

X ( jω) = ∫−∞

+∞

x (t) e− jω t dt



Continuous Time

Continuous Time Fourier Series

Continuous Time Fourier Transform

C k =1T∫0

Tx (t ) e− j k ω0 t dt x (t ) = ∑

k=−∞

+∞

Ck e+ j kω0 t

x (t ) =12π

∫−∞

+∞

X ( jω) e+ jω t dωX ( jω) = ∫−∞

+∞

x (t) e− jω t dt



Discrete Time

Discrete Time Fourier Series

Discrete Time Fourier Transform

γ[k ] =1N

∑n = 0

N−1

x [n] e− j k ω̂0n x [n] = ∑k = 0

N−1

γ[k ] e+ j k ω̂0n

x [n] =12π

∫−π

+π

X ( j ω̂) e+ j ω̂nd ω̂X ( j ω̂) = ∑n =−∞

+∞

x [n] e− j ω̂n



Continuous Time Signal Spectrum

CTFS


C kC−k

+ k ω0−k ω0

C k e+ j kω0 t

CTFT


X (+ jω)X (− jω)

+ ω−ω

X (+ jω)e+ jω tX (− jω)e− jω tC−k e− j kω 0 t

C k =1T∫0

Tx (t ) e− j k ω0 t dt X ( jω) = ∫

−∞

+∞

x (t) e− jω t dt



Discrete Time Signal Spectrum

DTFT


X (+ j ω̂)X (− j ω̂)

+ ω̂−ω̂

X (e+ j ω̂)e+ j ω̂ nX (e− j ω̂)e− j ω̂n

DTFS

only mag is display

γ [+k ]γ [−k ]

+k ω̂0−k ω̂0

γ [+k ] e+ jk ω̂0nγ [−k ]e− j k ω̂0n

+ π−π+ π−π

γ[k ] =1N

∑n = 0

N−1

x [n] e− j k ω̂0n X ( j ω̂) = ∑n =−∞

+∞

x [n] e− j ω̂n




References

[1] http://en.wikipedia.org/[2] J.H. McClellan, et al., Signal Processing First, Pearson Prentice Hall, 2003[3] M.J. Roberts, Fundamentals of Signals and Systems[4] S.J. Orfanidis, Introduction to Signal Processing[5] K. Shin, et al., Fundamentals of Signal Processing for Sound and Vibration Engineerings

[6] A “graphical interpretation” of the DFT and FFT, by Steve Mann

Documents

Spectrum Representation (5A) - Wikimedia€¦ · 28/10/2016 · 5A Spectrum Representation 5 Young Won Lim 10/28/16 Fourier Series with complex coefficients x(t) = ∑ k=−∞ C