analog communication

1. Introduction
• Engineers view signals in terms of frequency
spectra
• Audio signals having bandwidth of 20 kHz
• Loudspeakers responding to 20 kHz audio
signal
• We will study spectra representation of
aperiodic signals

2. Aperiodic Representation
• Aperiodic signal g(t) can be represented as a continuous sum
(integral) of everlasting exponential
• We have to construct a new periodic gT0(t) by repeating signal
g(t) every T0 seconds.
• gT0(t) can be represented as exponential Fourier series
• As T0 -> ∞ :
• Exponential Fourier series for gT0(t)

3. Aperiodic Representation
• Integrating gT0 over (-T0/2, T0/2) is the same as integrating g(t)
• If we define a function G(f)as a function ω
• Shows that Fourier Coefficient Dn are (1/T0 times) the samples of
G(f) uniformly spaced at intervals of f0 Hz
• (1/T0)(G(f)) is the envelope for the coefficient Dn
∫
∞
∞
−
−
= dt
e
t
g
f
G t
jω
)
(
)
( ∫
∞
∞
−
−
= dt
e
t
g ft
j π
2
)
(
)
(
1
0
0
nf
G
T
Dn =

4. Aperiodic Representation
• If T0 is doubled, the envelop is halved and magnitude
gets smaller as T0 is doubled more.
• Shape remains the same
• gT0(t) becomes
• Because T0 ->∞, f0 = 1/T0 becomes infinitesimal (f0->0).
Hence ∆f = 1/T0
• gT0(t) can be expressed as a sum of everlasting exponentials
of
t
f
jn
n
T e
T
nf
G
t
g 0
0
2
0
0 )
(
)
( π
∑
∞
−∞
=
=
t
f
n
j
n
T e
f
f
n
G
t
g )
2
(
]
)
(
[
)
(
0
∆
∞
−∞
=
∑ ∆
∆
= π
)
.....(
3
,
2
,
,
0 ies
fourierser
f
f
f ∆
±
∆
±
∆
±

5. • The amount of the component of frequency n∆f is
G(n∆f)∆f
• As T0 -> ∞, ∆f ->0 and gT0(t) ->g(t):
• This is the area under G(f)ej2πft called Fourier integral:
∑
∞
∞
−
∆
→
∆
∞
→
∆
∆
=
= f
e
f
n
G
t
gT
t
g t
f
n
j
f
T
)
2
(
0
0 )
(
)
(
)
( lim
lim
0
π
∫
∞
∞
−
= dt
e
f
G
t
g ft
j π
2
)
(
)
(

6. Fourier Tranform
• Direct Fourier transform of g(t)
– G(f)
• Inverse Fourier Transform of G(f)
• Symbolically:
• We can plot G(f) as a function of f and amplitude and
angle (phase) spectra exist
∫
∞
∞
−
−
= dt
e
t
g ft
j π
2
)
(
∫
∞
∞
−
= dt
e
f
G
t
g ft
j π
2
)
(
)
(
)]
(
[
)
(
)]
(
[
)
(
1
f
G
F
t
g
t
g
F
f
G
−
=
=

7. Conjugate Symmetry Property
• If g(t) is a real function of t, G(f) and G(-f) are complex
conjugate
G(-f) = G*(f)
• This means:
|G(-f)| = |G(f)|
θg(-f) = -θg(f)

8. Find the Fourier Transform of e-atu(t)
• hi
• Expressing a+jω in polar form

9. • The amplitude spectrum |G(f)| and the phase
spectrum θg(f) is shown:

10. Existence of Fourier Transform
• Fourier transform exist for a function g(t) if:
• The physical existence of a signal is a sufficient condition
for the existence of its transform
• The fourier transform is linear if:

11. Transform of Useful Functions
• Unit Rectangular Function
• The unit rectangular pulse rect(x) in (a) is expanded by a
factor τ rect(x/τ) in (b)

12. Unit Triangular function
• Triangular pulse ∆(x) of unit height and unit width,
centered at the origin:

13. Sinc function
• This is the function sin x/x (sine over argument)
• Plays an important role in signal processing
sinc(x) = sin x/x
• It is an even function of x
• Sinc (x) = 0 when sin x = 0 except at x = 0, i.e sinc (x) = 0
for t =
• Sinc(0) = 1
• Sinc (x) is an oscillating function with decreasing
amplitude.
• It has a unit peak at x = 0 and zero crossings at integer
multiples of π
• Product of sin x (T0 = 2π) and 1/x
π
π
π 3
,
2
, ±
±
±

14. Find the Fourier transform of g(t) = Π(t/τ)
• Solution
• Since Π(t/τ)=1 for |t| < τ/2, and since it is zero
for |t|> τ/2
dt
e
t
f
G ft
j π
τ
2
)
(
)
( −
∞
∞
−
∫ Π
=
dt
e
f
G ft
j π
τ
τ
2
2
2
)
( −
−
∫
=
)
(
sin
2
sin
)
(
sin
)
sin(
2
)
sin(
2
)
(
2
1
τ
π
τ
ωτ
τ
τ
τ
π
τ
τ
π
τ
π
τ
π
τ
π
π
τ
π
τ
π
f
c
c
t
f
c
f
f
f
f
e
e
f
j
f
j
f
j
=






⇔






Π
=
=
=
−
−
= −

15. • Since sinc(x) = 0 when x = ±nπ
• Sinc(ωt/2) = 0 when ωt/2 = ±nπ when f = ±n/τ (n = 1, 2 ,
3, ……)
• G(f) is real, hence spectra is just a single plot of G(f)
• See plot in the text
• The spectrum peaks at f = 0 and decays at higher
frequencies.
• Π(t/τ) is a low-pass signal with most of the signal energy
in lower frequency components
• Signal bandwidth – difference between highest
(significant) frequency and lowest (significant) frequency
in the signal spectrum
• A rough estimate of bandwidth of a rectangular pulse is
2π/τ rad/s or 1/τ Hz

16. Find the Fourier transform of the unit impulse signal (δ(t))
• Using sampling property of impulse function
∫
∞
∞
−
−
−
=
=
= 1
)
(
)]
(
[ 0
.
2
2 f
j
ft
j
e
dt
e
t
t
F π
π
δ
δ

17. Find the inverse Fourier transform of δ(2πf) = (1/2π)δ(f)
• Using sampling property of impulse function (pg. 33)
• The spectrum of a constant signal g(t) = 1 is an impulse
δ(f) = 2πδ(2πf)
• g(t) = 1 is a dc signal that has frequency f = 0(dc)
)
(
1
)
2
(
2
1
2
1
2
1
)
2
(
)
2
(
2
1
)
2
(
)]
2
(
[
.
0
2
2
1
f
f
e
f
d
e
f
df
e
f
f
F
t
j
ft
j
ft
j
δ
π
δ
π
π
π
π
π
δ
π
π
δ
π
δ π
π
⇔
⇔
=
=
=
=
−
∞
∞
−
∞
∞
−
−
∫
∫

18. Find the inverse Fourier transform of δ(f-f0)
• From sampling property of the impulse function
• This shows that the spectrum of an everlasting exponential
ej2πf0t is a single impulse at f = f0
• The spectrum is made up of a single component at
frequency f = f0
)
(
)
(
)]
(
[
0
2
2
2
0
0
1
0
0
f
f
e
e
df
e
f
f
f
f
F
t
f
j
t
f
j
ft
j
−
⇔
=
−
=
− ∫
∞
∞
−
−
δ
δ
δ
π
π
π
)
( 0
2 0
f
f
e t
f
+
⇔
−
δ
π

19. Find the Fourier transform of the everlasting sinusoid cos 2πf0t
• Using Euler formula
• The spectrum of cos 2πf0t consists of two impulses at f0
and –f0 in the f-domain or two impulses ±ω0 = ±2πf0 in
the ω –domain
• An everlasting sinusoid can be synthesized by two
everlasting exponentials ejω0t and ejω0t
• The fourier spectrum consists of only two components of
frequencies ω0 and -ω0
)]
(
)
(
[
2
1
2
cos
)
(
2
1
2
cos
0
0
0
2
2
0
0
0
f
f
f
f
t
f
e
e
t
f t
f
j
t
f
j
−
+
+
=
+
= −
δ
δ
π
π π
π

20. Find the Fourier transform of the sign function sgn(t) –(signum t)
its value is +1 or -1 depending on whether t is positive or
negative

23. Some properties of Fourier Transform
• Properties of Fourier transform, implications and
Applications will be studied
• Time-Frequency Duality- similar to photograph and its
negative
• If g(t) <=>G(f)
• Time-shifting property
• Dual of the time-shifting property
• There is role reversal, and we quarantee that any result
will have a dual

24. Duality Property
• If the fourier transform of g(t) is G(f) then the
fourier transform of G(t) with f replaced by t is
g(-f)
• g(-f) is the original time domain signal with t
replaced by -f

25. Apply duality property to the pair of figures below
• Figure
• G(t) is the same G(f), and g(-f) is the same as g(t) with t
replaced by –f
• Substituting τ = 2πα

26. Time-Scaling Property
• If
• For any real constant a
• Proof
• If a < 0
• A time compression of a signal results in a
spectral expansion and vice versa

27. Time scaling property
• Compression in time by a factor a means that the signal
is varying more rapidly by the same factor
• To synthesize, frequencies of the signal must be
increased by a (frequency spectrum expanded by a)
• A signal cos 4πf0t is the same as the signal cos 2πf0t time
compressed by a factor of 2

28. Reciprocity of signal and its bandwidth
• Time-scaling property implies that if g(t) is
wider, its spectrum is narrower and vice versa
• Doubling the signal duration halves it
bandwidth and vice versa
• Bandwidth of a signal is inversely proportional
to the signal duration or (width in seconds)
• E.g bandwidth of previous figures.

29. Example
• Show that
• Use this result and the fact that

30. Time-Shifting Property
• Delaying a signal by to second does not change
its amplitude spectrum, however the phase
spectrum is changed by -2πft0
• If
• Proof:
If t-t0 = x

31. Time-shifting Property
• Time delay in a signal causes linear phase shift
in its spectrum
• To achieve the same delay, higher frequency
sinusoids must undergo proportionately larger
phase shift

32. Example
• Find the Fourier transform
• Delay causes a linear phase spectrum

33. Frequency-Shifting Property
• Multiplication of a signal by a factor ejπfot shifts
the spectrum of the signal by f=f0
• If
Then
• Proof:
• Changing f to –f0

34. • Because ejπfot is not a real function in practise, frequency
shifting in practice is achieved by multiplying g(t) by a
sinusoid.
• Multiplication of g(t) by a sinusoid of frequency f0 shifts
the spectrum G(f) by ±f0.
• Mulitplication of a sinusoid cos 2πf0t by g(t) amounts to
modulating the sinusoid amplitude. (amplitude
modulation)
• cos 2πf0t is the carrier, g(t) is the modulating signal and
cos 2πf0t g(t) is the modulated signal (chapt 4 and 5)

35. • To sketch a signal g(t) cos 2πf0t
• g(t) cos 2πf0t touches g(t) when the sinusoid cos 2πf0t is
at its peaks and touches when cos 2πf0t is at its negative
peaks
• g(t) and –g(t) therefore acts as envelopes for the signal
g(t) cos 2πf0t
• g(t) and –g(t) are mirror images of each other about the
horizontal axis.

37. Shifting the phase spectrum of a
modulated signal
• By using cos (2πf0t+θ) instead of cos 2πf0t
• If signal g(t) is multiplied by cos (2πf0t+θ),
• If

38. Example
• Find and sketch the Fourier transform of the modulated
signal g(t)cos2πf0t in which g(t) is a rectangular pulse
Π(t/T) as shown below:
• From pair , G(f)
• Spectrum is shown below

39. Application of Modulation
• Modulation is used to shift signal spectra in this
scenarios:
– Interference will occur if signals occupying the same
frequency band are transmitted over the same medium
– Example, if radio stations broadcast audio signals, receiver
will not be able to separate them
– Radio stations could be assigned different carrier
frequency.
– Radio station transmit modulated signal, thus shifting the
signal spectrum to allocated band.
– Both modulation and demodulation utilizes spectra
shifting
– Transmitting several signals simultaneously over a channel
by using different frequency bands is called Frequency
division multiplexing

40. Application of Modulation
• Audio freqencies are so low (large wavelength) that it will
require impractical large antennas for radiation.
• Shifting the spectrum to a higher frequency (smaller
wavelength) by modulation will solve the problem