Signal Processing Lihi Zelnik-Manor #4 Fourier Transform of 1D signals

Outline for today • The Fourier Transform – Continuous signals • Periodic • Non-periodic

– Discrete signals

Periodic signals • A Fourier series represents a wave-like signal as a (possibly infinite) sum of sines and cosines (i.e., complex exponentials)

A simple periodic signal • A very simple periodic signal is a sine wave:

x(t )  a sin 0t • Complex signals can also be periodic:

x(t )  ae

j0t

A sum of periodic signals • Q: What will we get from a finite sum of periodic signals? N

x(t )   ak e

jk t

k 1

• Ans: This is also a periodic signal • Note: We are only using frequencies 𝜔𝑘 = 𝑘𝜔0 where 𝜔0 is a constant

An infinite sum of periodic signals • Q: What will we get from an infinite sum of periodic signals? 

x(t )   ak e

jk t

k 1

• Ans: It turns out that if





a   k k 1

then 𝑥(𝑡) is well defined

• Such signals have more than one frequency

Fourier series • A periodic signal with period 𝑇0 can be described mathematically like this: 𝑥(𝑡 + 𝑛𝑇0 ) = 𝑥(𝑡) • Fourier showed that any complex periodic signal can be decomposed as: ∞

𝑎𝑘 𝑒 𝑗𝑘𝜔0𝑡

𝑥(𝑡) = 𝑘=−∞

Where 𝜔0 = 2 𝜋 𝑇0 and 1 𝑎𝑘 = 2𝜋

𝜋

𝑥(𝑡)𝑒 −𝑗𝑘𝜔0 𝑡 𝑑𝑡 −𝜋

Triangular wave 1 coefficients

The Fourier series for a triangular wave is: ( 1)( n 1)/2  n t  x(t )  2  sin   2  n 1,3,5... n  L  8



2 coefficients

1

1

0.5

0.5

0

0

-0.5

-0.5

-1 -2

-1

0

1

2

-1 -2

5 coefficients 1

0.5

0.5

0

0

-0.5

-0.5

-1

0

0

1

2

1

2

20 coefficients

1

-1 -2

-1

1

2

-1 -2

-1

0

Error Graph MSE 

1 M



f ( x)   triangular  𝑥(𝑡)

Infinite _ error  max  triangular  𝑥(𝑡) f ( x) 

2

period

period

Infinite error Vs. coefficients number

Mean square error Vs. coefficients number -3

5

x 10

0.2

4.5

0.18

4

0.16

3.5

0.14

3

0.12

2.5

0.1

2

0.08

1.5

0.06

1

0.04

0.5

0.02

0

0 0

20

40

60

80

100

0

20

40

60

80

100

Rectangular Wave The Fourier series for a square wave is: 1  n t  x(t )  sin    n 1,3,5... n  L  4



5 coefficients

2 coefficients

1 coefficients 2

2

2

1

1

1

0

0

0

-1

-1

-1

-2 -2

0

2

-2 -2

0

2

-2 -2

2

2

2

1

1

1

0

0

0

-1

-1

-1

-2 -2

0

2

-2 -2

0

2

500 coefficients

50 coefficients

15 coefficients

0

2

-2 -2

0

2

Error Graph 1 MSE  M

f ( x)    SquareWave  𝑥(𝑡)

2

Infinite _ error  max  SquareWave 𝑥(𝑡) f ( x)  period

period Mean square error Vs. number of coefficients

Infinite error Vs. coefficients number

0.2

1

0.18

0.9

0.16

0.8

0.14

0.7

0.12

0.6

0.1

0.5

0.08

0.4

0.06

0.3

0.04

0.2

0.02

0.1

0

0

500

1000

0

0

500

1000

Fourier series • A periodic signal with period 𝑇0 can be described mathematically like this: 𝑥(𝑡 + 𝑛𝑇0 ) = 𝑥(𝑡) • Fourier showed that any complex periodic signal can be decomposed as: ∞

𝑎𝑘 𝑒 𝑗𝑘𝜔0𝑡

𝑥(𝑡) = 𝑘=−∞

Where 𝜔0 = 2 𝜋 𝑇0 and 1 𝑎𝑘 = 2𝜋

𝜋

𝑥(𝑡)𝑒 −𝑗𝑘𝜔0 𝑡 𝑑𝑡 −𝜋

What are the frequencies? • Q: How can we tell what frequencies 𝜔𝑘 = 𝑘𝜔0 a signal has?

• We can easily visualize the frequencies via the following continuous function: ∞ a0 A( ) a 1 a2 𝐴(𝜔) = 2𝜋 𝑎𝑘 𝛿(𝜔 − 𝑘𝜔0 ) a 1 a 𝑘=−∞

2



What are the frequencies? • Using some mathematical tricks we can re-write our signal as: 1 𝑥(𝑡) = 2𝜋

∞

𝐴(𝜔)𝑒 𝑗𝜔𝑡 𝑑𝜔 −∞

• We could do this because 𝐴(𝜔) is a sum of deltas

What are the frequencies? 𝑥(𝑡)

𝐴(𝜔)

Non-periodic signals • The Fourier Integral represents non-periodic signals as a “sum” of periodic signals (complex exponents): 1 𝑥(𝑡) = 2𝜋

∞

X(𝜔)𝑒 𝑗𝜔𝑡 𝑑𝜔 −∞

• The Fourier Transform defines the coefficients X(𝜔): ∞

𝑥(𝑡)𝑒 −𝑗𝜔𝑡 𝑑𝑡

X 𝜔 = −∞

Some important notes • The Fourier Integral and the Fourier transform are not always the inverse of each other

• When both 𝑥(𝑡) and X 𝜔 are absolutely integrable then the inversion holds. (but also it could work for other functions)

Fourier examples Temporal domain cos(2𝜋𝑎𝑡) 𝑡

𝜔

𝑡

𝜔

cos(2𝜋𝑏𝑡)

Fourier examples Temporal domain Π(𝑡) 𝜔

𝑡

G𝑎𝑢𝑠𝑠(𝑡, 𝜎)

G𝑎𝑢𝑠𝑠(𝜔, 1/𝜎) 𝑡

G𝑎𝑢𝑠𝑠(𝑡, 1/𝜎) 𝑡

𝜔

G𝑎𝑢𝑠𝑠(𝜔, 𝜎) 𝜔

Fourier transform example 𝑋(𝜔) 𝑥(𝑡)

arg{𝑋 𝜔 }

Fourier Transforms

Fourier Series (FS) Fourier Transform (FT)

Time Continuous, Periodic Continuous, Infinite

Frequency Discrete, Infinite Continuous, Infinite

Discrete-Time FT (DTFT) Discrete FT (DFT)

Discrete, Infinite Discrete, Finite / periodic

Continuous, Periodic Discrete, Finite / periodic

Discrete Time FT (DTFT) • For any discrete signal we define its DTFT as: ∞

𝑥[𝑛]𝑒 −𝑗𝑛𝛺

𝑋(𝛺) = 𝑛=−∞

• 𝛺 is measured in radians. • We can relate it to frequency 𝑓 in Hz by: 𝛺 = 2𝜋/𝑓 • The DTFT is defined only when ∞

|𝑥[𝑛]| < ∞ 𝑛=−∞

DTFT example • Lets take for example a right-hand exponent 𝑥 𝑛 = • If we’ll compute its DTFT we’ll get 𝑋 𝛺 = • We can plot this:

𝑋 Ω • You can see it is periodic

1 𝑛 𝛼

𝑛≥0

1 1−𝛼𝑒 −𝑗𝜔

arg{𝑋 Ω }

𝑋(Ω)

Discrete Time FT (DTFT) • DTFT is always periodic: 𝑋 𝛺 = 𝑋 𝛺 + 2𝜋𝑘 • This is true since 𝑒 −𝑗𝑛2𝜋𝑘 = 1 for any integers 𝑘, 𝑛

• This means that knowing 𝑋 𝛺 in the interval 𝛺 = [−𝜋, 𝜋] suffices

DTFT and FT

Sampling audio signals • Q: The standard CD recording sampling rate is 𝑓𝑠𝑎𝑚 =44.1kHz. Why? • Ans: – The range [−𝜋, 𝜋] for 𝛺 corresponds to the range [−0.5𝑓𝑠𝑎𝑚 , 0.5𝑓𝑠𝑎𝑚 ] in kHz – For CD’s this means [−22.05, +22.05]kHz – Our ears can only hear up to ~20kHz, so 22.05kHz suffices Recall: 𝛺 = 2𝜋/𝑓

Inverse Discrete Time FT (DTFT) • (sometimes) We can obtain a signal from its DTFT: 1 𝜋 𝑥𝑛 = 𝑋(𝛺)𝑒 𝑗𝛺𝑛 𝑑𝛺 2𝜋 −𝜋 • This is actually a Fourier series analysis

Time-Frequency scale • Narrow in time  wide in frequency 𝑋(𝛺)

𝛺

• And vice versa for 𝑥 𝑛 = 1 we get 𝑋(𝛺) =2𝜋δ(𝛺)

DTFT of real x[n] • Magnitude is symmetric • Phase is anti-symmetric

Fourier Transforms

Fourier Series (FS) Fourier Transform (FT)

Time Continuous, Periodic Continuous, Infinite

Frequency Discrete, Infinite Continuous, Infinite

Discrete-Time FT (DTFT) Discrete FT (DFT)

Discrete, Infinite Discrete, Finite / periodic

Continuous, Periodic Discrete, Finite / periodic

Discrete FT (DFT) • A finite or periodic discrete signal has only N unique values • Its spectrum can be defined by N discrete frequency samples 1 𝑥[𝑛] = 𝑁

𝑁−1

𝑋[𝑘]𝑒 𝑗𝑛𝑘𝛺0 𝑘=0

𝑁−1

𝑥[𝑛]𝑒 −𝑗𝑛𝑘𝛺0

𝑋[𝑘] = 𝑛=0

DFT and DTFT • DFT

𝑋[𝑘] =

• DTFT

𝑋(𝛺) =

𝑁−1 −𝑗𝑛𝑘𝛺0 𝑥[𝑛]𝑒 𝑛=0 ∞ −𝑗𝑛𝛺 𝑥[𝑛]𝑒 𝑛=−∞

• DFT “samples” DTFT at discrete frequencies

𝑋(𝛺)

𝛺 = 𝑘𝛺0

𝑋(𝛺) 𝛺

DFT and DTFT

DFT, DTFT and FT

Some cool Fourier facts • Energy (sum of squares) is the same in each domain – Rayleigh’s theorem (Parseval)

• Temporal shift only changes phase • Transform of a Gaussian is a Gaussian • We’ll see this once more, after we do convolution

• In a conventional DSP course these are homework exercises

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