Signals and Systems: Part 2 - Stanford...

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Signals and Systems: Part 2 The Fourier transform in 2πf Some important Fourier transforms Some important Fourier transform theorems Convolution and Modulation Ideal filters

Transcript of Signals and Systems: Part 2 - Stanford...

Page 1: Signals and Systems: Part 2 - Stanford Universityweb.stanford.edu/class/ee179/lectures/notes04.pdfSignals and Systems: Part 2 The Fourier transform in 2πf Some important Fourier transforms

Signals and Systems: Part 2

◮ The Fourier transform in 2πf

◮ Some important Fourier transforms

◮ Some important Fourier transform theorems

◮ Convolution and Modulation

◮ Ideal filters

Page 2: Signals and Systems: Part 2 - Stanford Universityweb.stanford.edu/class/ee179/lectures/notes04.pdfSignals and Systems: Part 2 The Fourier transform in 2πf Some important Fourier transforms

Fourier transform definitions

◮ In EE 102A, the Fourier transform and its inverse were defined by

G(jω) =

∫ ∞

−∞g(t)e−jωt dt , g(t) =

1

∫ ∞

−∞G(jω)ejωt dω

We used jω to make all transforms similar in form (Fourier, Laplace,DTFT, z).

◮ In this class we use 2πf instead of ω. If we replace ω with 2πf ,

G(f) =

∫ ∞

−∞g(t)e−j2πft dt , g(t) =

∫ ∞

−∞G(f)ej2πft df

The transform and its inverse are almost symmetric; only the sign in thecomplex exponential changes.

◮ This will simplify a lot of the transforms and theorems.

Page 3: Signals and Systems: Part 2 - Stanford Universityweb.stanford.edu/class/ee179/lectures/notes04.pdfSignals and Systems: Part 2 The Fourier transform in 2πf Some important Fourier transforms

Fourier transform existence

◮ If g(t) is absolutely integrable, i.e.,∫ ∞

−∞|g(t)| dt < ∞

then G(f) exists for every frequency f and is continuous.

◮ If g(t) has finite energy, i.e.,∫ ∞

−∞|g(t)|2 dt < ∞

then G(f) exists for “most” frequencies f and has finite energy.

◮ If g(t) is periodic and has a Fourier series, then

G(f) =∞∑

n=−∞

G(nf0)δ(f − nf0)

is a weighted sum of impulses in frequency domain.

Page 4: Signals and Systems: Part 2 - Stanford Universityweb.stanford.edu/class/ee179/lectures/notes04.pdfSignals and Systems: Part 2 The Fourier transform in 2πf Some important Fourier transforms

Fourier transform examples

One-sided exponential decay is defined by e−atu(t) with a > 0:

g(t) =

{

0 t < 0

e−at t > 0

The Fourier transform of one-sided decay is (simple calculus):

G(f) =

∫ ∞

0e−at e−j2πft dt

=

∫ ∞

0e−(a+j2πf)t dt

=

[

e−(a+j2πf)t

−(a+ j2πf)

]t=∞

t=0

=1

a+ j2πf

Since g(t) has finite area, its transform is continuous.

Even though g(t) is real, its transform is complex valued. This is the usualsituation.

Page 5: Signals and Systems: Part 2 - Stanford Universityweb.stanford.edu/class/ee179/lectures/notes04.pdfSignals and Systems: Part 2 The Fourier transform in 2πf Some important Fourier transforms

Fourier transform examples (cont.)

We can rationalize G(f):

G(f) =1

a+ j2πf=

a− j2πf

a2 + 4π2f2=

a

a2 + 4π2f2− j

2πf

a2 + 4π2f2

We can better picture G(f) using polar representation.

|G(f)| =1

a2 + 4π2f2, θG(f) = ∠G = − tan−1

(−2πf

a

)

Page 6: Signals and Systems: Part 2 - Stanford Universityweb.stanford.edu/class/ee179/lectures/notes04.pdfSignals and Systems: Part 2 The Fourier transform in 2πf Some important Fourier transforms

Fourier transform examples (cont.)

Fourier transform at f = 0:

G(0) =

∫ ∞

−∞g(t)e−j2π·0·t dt =

∫ ∞

−∞g(t) dt

In general, G(0) is the area under g(t), called the DC value.

For g(t) = e−atu(t), G(0) = 1/a is the only real value. It is also largest inmagnitude.

For one-sided decay, G(−f) = G∗(f), complex conjugate of G(f).This is true for all real-valued signals.

G∗(f) =(

∫ ∞

−∞g(t)e−j2πft dt

)∗=

∫ ∞

−∞g(t)∗(e−j2πft)∗ dt

=

∫ ∞

−∞g(t)ej2πft dt =

∫ ∞

−∞g(t)e−j2π(−f)t dt = G(−f)

We need look at only positive frequencies for real-valued signals.

Page 7: Signals and Systems: Part 2 - Stanford Universityweb.stanford.edu/class/ee179/lectures/notes04.pdfSignals and Systems: Part 2 The Fourier transform in 2πf Some important Fourier transforms

Duality

Fourier inversion theorem: if g(t) has Fourier transform G(f), then

G(f) =

∫ ∞

−∞g(t)e−j2πft dt and g(t) =

∫ ∞

−∞G(f)ej2πft df

Inverse transform differs from forward transform only in sign of exponent.

Suppose we consider G(·) to be a function of t instead of f .If we apply the Fourier transform again:

∫ ∞

−∞G(t)e−j2πft dt =

∫ ∞

−∞G(t)ej2π(−f)t dt = g(−f)

We can summarize this as

F{g(t)} = G(f) ⇒ F{G(t)} = g(−f)

This is the principle of duality.

Page 8: Signals and Systems: Part 2 - Stanford Universityweb.stanford.edu/class/ee179/lectures/notes04.pdfSignals and Systems: Part 2 The Fourier transform in 2πf Some important Fourier transforms

Important Fourier transforms

The unit rectangle function Π(t) is defined by

Π(t) =

{

1 |t| < 12

0 |t| > 12

Its Fourier transform is

F {Π(t)} =

∫ ∞

−∞e−i2πftΠ(t) dt

=

∫ 1/2

−1/2e−i2πft dt =

sinπf

πf= sinc(πf)

By duality, we know that

F{sinc(πt)} = Π(−f) = Π(f)

since Π(f) is even. We say Π(t) and sinc(πf) are a Fourier pair :

Π(t) ⇋ sinc(πf)

Fact: every finite width pulse has a transform with unbounded frequencies.

Page 9: Signals and Systems: Part 2 - Stanford Universityweb.stanford.edu/class/ee179/lectures/notes04.pdfSignals and Systems: Part 2 The Fourier transform in 2πf Some important Fourier transforms

Fourier transform time scaling

If a > 0 and g(t) is a signal with Fourier transform G(f), then

F {g(at)} =

∫ ∞

−∞g(at)e−j2πftdt =

∫ ∞

−∞g(u)e−j2πf(u/a) du

a=

1

aG(f

a

)

If a < 0, change of variables requires reversing limits of integration:

F {g(at)} = −1

aG(f

a

)

Combining both cases:

F {g(at)} =1

|a|G(f

a

)

Special case: a = −1. The Fourier transform of g(−t) is G(−f).

Compressing in time corresponds to expansion in frequency (and reductionin amplitude) and vice versa.

The sharper the pulse the wider the spectrum.

Page 10: Signals and Systems: Part 2 - Stanford Universityweb.stanford.edu/class/ee179/lectures/notes04.pdfSignals and Systems: Part 2 The Fourier transform in 2πf Some important Fourier transforms

Fourier transform time scaling example

The transform of a narrow rectangular pulse of area 1 is

F{1

τΠ(t/τ)

}

= sinc(πτf)

In the limit, the pulse is the unit impulse, and its tranform is the constant 1.

We can find the Fourier transform directly:

F{δ(t)} =

∫ ∞

−∞δ(t)e−j2πft dt = e−j2πft

t=0= 1

soδ(t) ⇋ 1

The impulse is the mathematical abstraction of signal whose Fouriertransform has magnitude 1 and phase 0 for all frequencies.

By duality, F{1} = δ(f). All DC, no oscillation.

1 ⇋ δ(f)

Page 11: Signals and Systems: Part 2 - Stanford Universityweb.stanford.edu/class/ee179/lectures/notes04.pdfSignals and Systems: Part 2 The Fourier transform in 2πf Some important Fourier transforms

Important Fourier transforms (cont.)

◮ Shifted impulse δ(t− t0):

F{δ(t − t0)} =

∫ ∞

−∞δ(t − t0)e

−j2πft dt = e−j2πft0

This is a complex exponential in frequency.

δ(t− t0) ⇋ e−j2πft0

This is an example of shift theorem: F{f(t− t0)} = e−j2πft0F{f(t)}.

By duality, complex exponential in time has an impulse in frequency:

ej2πf0t ⇋ δ(f − f0)

◮ Sinuoids: frequency content is concentrated at ±f0 Hz.

F{cos 2πf0t} = F{12 (e

j2πf0t + e−j2πf0t)}

= 12δ(f − f0) +

12δ(f + f0)

F{sin 2πf0t} = F{ 12i (e

j2πf0t + e−j2πf0t)}

= 12iδ(f − f0)−

12iδ(f + f0)

Page 12: Signals and Systems: Part 2 - Stanford Universityweb.stanford.edu/class/ee179/lectures/notes04.pdfSignals and Systems: Part 2 The Fourier transform in 2πf Some important Fourier transforms

Important Fourier transforms (cont.)

◮ Laplacian pulse g(t) = e−a|t| where a > 0. Since

g(t) = e−atu(t) + eatu(−t) ,

we can use reversal and additivity:

G(f) =1

a+ j2πf+

1

a− j2πf=

2a

a2 + 4π2f2.

This is twice the real part of the Fourier transform of e−atu(t)

◮ The signum function can be approximated as

sgn(t) = lima→0

(

e−atu(t)− eatu(−t))

This has the Fourier transform

F {sgn(t)} = lima→0

(

1

a+ j2πf−

1

a− j2πf

)

=1

jπf

Page 13: Signals and Systems: Part 2 - Stanford Universityweb.stanford.edu/class/ee179/lectures/notes04.pdfSignals and Systems: Part 2 The Fourier transform in 2πf Some important Fourier transforms

Important Fourier transforms (cont.)

◮ The unit step function is

u(t) = 12 +

12sgn(t)

This has the Fourier transform

F{u(t)} = 12δ(f) +

1

j2πf

Page 14: Signals and Systems: Part 2 - Stanford Universityweb.stanford.edu/class/ee179/lectures/notes04.pdfSignals and Systems: Part 2 The Fourier transform in 2πf Some important Fourier transforms

Fourier transform properties

◮ Time delay causes linear phase shift.

F {g(t− t0)} =

∫ ∞

−∞g(t− t0)e

−j2πft dt

=

∫ ∞

−∞g(u)e−j2πf(u+t0) du

= e−j2πft0

∫ ∞

−∞g(u)e−j2πfu du = e−j2πft0G(f)

Thereforeg(t− t0) ⇋ e−j2πft0G(f)

◮ By duality we get frequency shifting (modulation):

ej2πfctg(t) ⇋ G(f − fc)

cos(2πfct)g(t) ⇋12G(f + fc) +

12G(f − fc)

Page 15: Signals and Systems: Part 2 - Stanford Universityweb.stanford.edu/class/ee179/lectures/notes04.pdfSignals and Systems: Part 2 The Fourier transform in 2πf Some important Fourier transforms

Fourier transform properties (cont.)

◮ Convolution in time. The convolution of two signals is

g1(t) ∗ g2(t) =

∫ ∞

−∞g1(u)g2(t− u) du

◮ The Fourier transform of the convolution is the product of thetransforms.

g1(t) ∗ g2(t) ⇋ G1(f)G2(f)

The Fourier transform reduces convolution to a simpler operation.

◮ Note that there is no factor of12π for frequency domain convolution,

as there was in EE 102A.

Page 16: Signals and Systems: Part 2 - Stanford Universityweb.stanford.edu/class/ee179/lectures/notes04.pdfSignals and Systems: Part 2 The Fourier transform in 2πf Some important Fourier transforms

Fourier transform properties (cont.)

◮ Multiplication in time.

g1(t)g2(t) ⇋

∫ ∞

−∞G1(λ)G2(f − λ) dλ

◮ Again, there is no12π factor, as there was in EE 102A.

◮ Convolution in one domain goes exactly to multiplication in the otherdomain, and multiplication to convolution.

◮ The modulation theorem is a special case.

F {g(t) cos(2πfct)} = F {g(t))} ∗ F {cos(2πfct)}

= G(f) ∗(

12δ(f + f0) +

12δ(f − f0)

)

= 12G(f + fc) + 1

2G(f − fc)

Page 17: Signals and Systems: Part 2 - Stanford Universityweb.stanford.edu/class/ee179/lectures/notes04.pdfSignals and Systems: Part 2 The Fourier transform in 2πf Some important Fourier transforms

The triangle function ∆(t) and its Fourier transform

◮ The book defines the triangle function ∆(t) as

∆(t) =

{

1− 2|x| |x| ≤ 12

0 otherwise

This is another unfortunate choice, but not as bad as sinc(t)!

◮ The triangle function can be written as twice the convolution of tworectangle functions of width 1

2 .

∆(t) = 2 Π(2t) ∗ Π(2t)

where the factor of 2 is needed to make the convolution 1 at t = 0.

◮ The Fourier transform is therefore

F {∆(t)} = F {2Π(2t) ∗Π(2t)} = 2F(

{Π(2t)})2

= 2(

12 sinc

(

π2 f

))2= 1

2 sinc2(

12πf

)

Then∆(t) ⇋ 1

2 sinc2(12πf)

Page 18: Signals and Systems: Part 2 - Stanford Universityweb.stanford.edu/class/ee179/lectures/notes04.pdfSignals and Systems: Part 2 The Fourier transform in 2πf Some important Fourier transforms

Modulation theorem

Modulation of a baseband signal creates replicas at ± the modulationfrequency.

Page 19: Signals and Systems: Part 2 - Stanford Universityweb.stanford.edu/class/ee179/lectures/notes04.pdfSignals and Systems: Part 2 The Fourier transform in 2πf Some important Fourier transforms

Applications of modulation

◮ For transmission by radio, antenna size is proportional to wavelength.

Low frequency signals (voice, music) must be converted to higherfrequency.

◮ To share bandwidth, signals are modulated by different carrierfrequencies.

◮ North America AM radio band: 535–1605 KHz (10 KHz bands)

◮ North America FM radio band: 88–108 MHz (200 KHz bands)

◮ North America TV bands: VHF 54–72, 76–88, 174–216, UHF 470–806,806–890

Frequencies can be reused in different geographical areas.

With digital TV, channel numbers do not correspond to frequencies.

Rats laughing: http://www.youtube.com/watch?v=j-admRGFVNM

Page 20: Signals and Systems: Part 2 - Stanford Universityweb.stanford.edu/class/ee179/lectures/notes04.pdfSignals and Systems: Part 2 The Fourier transform in 2πf Some important Fourier transforms

Bandpass signals

Bandlimited signal: G(f) = 0 if |f | > B.

Every sinusoid sin(2πfct) has bandwidth fc.

If gc(t) and gs(t) are bandlimited, then

m(t) = gc(t) cos(2πfct) + gs(t) sin(2πfct)

is a bandpass signal. Its Fourier transform or spectrum is restricted to

fc −B < |f | < fc +B

The bandwidth is (fc +B)− (fc −B) = 2B.

Most signals of interest in communications will be either bandpass (RF), orbaseband (ethernet).

Page 21: Signals and Systems: Part 2 - Stanford Universityweb.stanford.edu/class/ee179/lectures/notes04.pdfSignals and Systems: Part 2 The Fourier transform in 2πf Some important Fourier transforms

Filters

A filter is a system that modifies an input. (E.g., an optical filter blockscertain frequencies of light.)

In communication theory, we usually consider linear filters, which are lineartime-invariant systems.

L(v1(t) + v2(t)) = L(v1(t)) + L(v2(t))

L(av(t)) = aL(v(t))

L(v(t− t0)) = (Lv)(t − t0)

Fundamental fact: every LTIS is defined by convolution:

Lv(t) = h(t) ∗ v(t) =

∫ ∞

−∞h(u)v(t− u) du =

∫ ∞

−∞h(t− u)v(u) du

The signal h(t) is called the impulse response because

Lδ(t) =

∫ ∞

−∞δ(u)h(t − u) du = h(t)

Page 22: Signals and Systems: Part 2 - Stanford Universityweb.stanford.edu/class/ee179/lectures/notes04.pdfSignals and Systems: Part 2 The Fourier transform in 2πf Some important Fourier transforms

Transfer function

By the convolution theorem,

h(t) ∗ v(t) ⇋ H(f)V (f)

H(f) is called the transfer function.

Many systems are best understood in the frequency domain.

Example: low pass filter with cutoff frequency B:

H(f) = Π( f

2B

)

e−j2πftd ⇋ 2B sinc(2πB(t− td))

Note that the system is not (cannot) be causal.

Page 23: Signals and Systems: Part 2 - Stanford Universityweb.stanford.edu/class/ee179/lectures/notes04.pdfSignals and Systems: Part 2 The Fourier transform in 2πf Some important Fourier transforms

Low-pass filter examplex(t) = cos(2π(0.3)t)+ cos(2π(0.8)t), h(t) = sinc(πt), y(t) = cos(2π(0.3)t)

−10 −8 −6 −4 −2 0 2 4 6 8 10−2

−1

0

1

2

−10 −8 −6 −4 −2 0 2 4 6 8 10

−1

−0.5

0

0.5

1

Page 24: Signals and Systems: Part 2 - Stanford Universityweb.stanford.edu/class/ee179/lectures/notes04.pdfSignals and Systems: Part 2 The Fourier transform in 2πf Some important Fourier transforms

Butterworth filter: nonideal low-pass filter

Butterworth filter has |H(f)| =1

(f/B)2n

Page 25: Signals and Systems: Part 2 - Stanford Universityweb.stanford.edu/class/ee179/lectures/notes04.pdfSignals and Systems: Part 2 The Fourier transform in 2πf Some important Fourier transforms

Butterworth filter vs. ideal low-pass filter

Page 26: Signals and Systems: Part 2 - Stanford Universityweb.stanford.edu/class/ee179/lectures/notes04.pdfSignals and Systems: Part 2 The Fourier transform in 2πf Some important Fourier transforms

High-pass and band-pass filters

If the phase is zero

hhighpass(t) = δ(t)− 2B sinc(2πBt)

hbandpass(t) = 2 cos(2πf0t) 2B sinc(2πBt)

In practice there will be an additional linear phase in the spectral profile,corresponding to a delay in time. This makes the filters realizable.

Page 27: Signals and Systems: Part 2 - Stanford Universityweb.stanford.edu/class/ee179/lectures/notes04.pdfSignals and Systems: Part 2 The Fourier transform in 2πf Some important Fourier transforms

Band-pass filter example

x(t) = cos(2π(0.3)t) + cos(2π(0.8)t), h(t) = cos 2πt sincπt,y(t) = cos(2π(0.8)t)

−10 −8 −6 −4 −2 0 2 4 6 8 10−2

−1

0

1

2

−10 −8 −6 −4 −2 0 2 4 6 8 10

−1

−0.5

0

0.5

1

Page 28: Signals and Systems: Part 2 - Stanford Universityweb.stanford.edu/class/ee179/lectures/notes04.pdfSignals and Systems: Part 2 The Fourier transform in 2πf Some important Fourier transforms

Low-pass filter: x(t) = Π( t4), h(t) = sinc t

−10 −8 −6 −4 −2 0 2 4 6 8 10

0

0.2

0.4

0.6

0.8

1

−10 −8 −6 −4 −2 0 2 4 6 8 10

−0.2

0

0.2

0.4

0.6

0.8

1

Page 29: Signals and Systems: Part 2 - Stanford Universityweb.stanford.edu/class/ee179/lectures/notes04.pdfSignals and Systems: Part 2 The Fourier transform in 2πf Some important Fourier transforms

Low-pass filter: x(t) = Π( t4), h(t) = sinc t

−10 −8 −6 −4 −2 0 2 4 6 8 10

0

0.2

0.4

0.6

0.8

1

−10 −8 −6 −4 −2 0 2 4 6 8 10−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Page 30: Signals and Systems: Part 2 - Stanford Universityweb.stanford.edu/class/ee179/lectures/notes04.pdfSignals and Systems: Part 2 The Fourier transform in 2πf Some important Fourier transforms

Band-pass filter: x(t) = Π( t4), h(t) = cos 2πt sinc t

−10 −8 −6 −4 −2 0 2 4 6 8 10

0

0.2

0.4

0.6

0.8

1

−10 −8 −6 −4 −2 0 2 4 6 8 10

−1

−0.5

0

0.5

1

Page 31: Signals and Systems: Part 2 - Stanford Universityweb.stanford.edu/class/ee179/lectures/notes04.pdfSignals and Systems: Part 2 The Fourier transform in 2πf Some important Fourier transforms

Examples of Communication Channels

◮ wires (PCD trace or conductor on IC)

◮ optical fiber (attenuation 4dB/km)

◮ broadcast TV (50 kW transmit)

◮ voice telephone line (under -9 dbm or 110 µW)

◮ walkie-talkie: 500 mW, 467 MHz

◮ Bluetooth: 20 dBm, 4 dBm, 0 dBm

◮ Voyager: X band transmitter, 160 bit/s, 23 W, 34m dish antenna

http://science.time.com/2013/03/20/humanity-leaves-the-solar-system

-35-years-later-voyager-offically-exits-the-heliosphere/

Page 32: Signals and Systems: Part 2 - Stanford Universityweb.stanford.edu/class/ee179/lectures/notes04.pdfSignals and Systems: Part 2 The Fourier transform in 2πf Some important Fourier transforms

Communication Channel Distortion

The linear description of a channel is its impulse response h(t) orequivalently its transfer function H(f).

y(t) = h(t) ∗ x(t) ⇋ Y (f) = H(f)X(f)

Note that H(f) both attenuates (|H(f)|) and phase shifts (∠H(f)) theinput signal.

Channels are subject to impairments:

◮ Nonlinear distortion (e.g., clipping)

◮ Random noise (independent or signal dependent)

◮ Interference from other transmitters

◮ Self interference (reflections or multipath)

Page 33: Signals and Systems: Part 2 - Stanford Universityweb.stanford.edu/class/ee179/lectures/notes04.pdfSignals and Systems: Part 2 The Fourier transform in 2πf Some important Fourier transforms

Channel Equalization

Linear distortion can be compensated for by equalization.

Heq(f) =1

H(f)⇒ X̂(f) = Heq(f)Y (f) = X(f)

The equalization filter accentuates frequencies that are attenuated by thechannel.

However, if y(t) includes noise or interference,

y(t) = x(t) + z(t)

then

Heq(f)Y (f) = X(f) +Z(f)

H(f)

Equalization may accentuate noise!

Page 34: Signals and Systems: Part 2 - Stanford Universityweb.stanford.edu/class/ee179/lectures/notes04.pdfSignals and Systems: Part 2 The Fourier transform in 2πf Some important Fourier transforms

Next time

◮ Lab this Friday : How do you figure out where signals are?Spectrograms and waterfall plots

◮ Next class Monday : 3.6 – 3.8 in Lathi and Ding. Signal distortion,power spectral density, correlation and autocorrelation.

◮ Wednesday : Begin Chapter 4. Analog modulation schemes.

◮ Lab next Friday : finding and decoding airband AM