6.02 Fall 2012
Lecture #14
•
6.02 Fall 2012
Spectral content via the DTFT
Lecture 14 Slide #1
Demo: “Deconvolving” Output of
Channel with Echo
x[n]
Channel,
h1[.]
y[n]
Receiver
filter, h2[.]
z[n]
Suppose channel is LTI with
h 1[n]=δ[n]+0.8δ[n-1]
− jΩm
h
[m]e
H1(Ω) = ?? = ∑ 1
m
So:
= 1+ 0.8e–jΩ = 1 + 0.8cos(Ω) – j0.8sin(Ω)
|H1(Ω)| = [1.64 + 1.6cos(Ω)]1/2
EVEN function of Ω;
<H1(Ω) = arctan [–(0.8sin(Ω)/[1 + 0.8cos(Ω)]
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ODD .
Lecture 14 Slide #2
A Frequency-Domain view of Deconvolution
x[n]
Channel,
H 1(Ω)
y[n]
Receiver
filter, H2(Ω)
z[n]
Noise w[n]
Given H1(Ω), what should H2(Ω) be, to get z[n]=x[n]?
H2(Ω)=1/H1(Ω)
“Inverse filter”
= (1/|H1(Ω)|). exp{–j<H1(Ω)}
Inverse filter at receiver does very badly in the presence of noise
that adds to y[n]:
filter has high gain for noise precisely at frequencies where
channel gain|H 1(Ω)| is low (and channel output is weak)!
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Lecture 14 Slide #3
DT Fourier Transform (DTFT) for
Spectral Representation of General x[n]
If we can write
1
h[n] =
2π
∫
H (Ω)e jΩn dΩ
<2 π >
∫
<2 π >
X(Ω)e
H (Ω) = ∑ h[m]e− jΩm
Any contiguous
interval of length
2
then we can write
1
x[n] =
2π
where
jΩn
dΩ
where
m
X(Ω) = ∑ x[m]e− jΩm
m
This Fourier representation expresses x[n] as
a weighted combination of e jΩn for all Ω in [–,].
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X(Ωο)dΩ is the spectral content of x[n]
in the frequency interval [Ωο, Ωο+ dΩ ]
Lecture 14 Slide #4
The spectrum of the exponential signal (0.5)nu[n] is shown over the
frequency range Ω = 2πf in [-4π,4π], The angle has units of degrees.
Courtesy of Don Johnson. Used with permission; available under a CC-BY license.
6.02 Fall 2012
http://cnx.org/content/m0524/latest/
Lecture 14 Slide #5
x[n] and X(Ω
Ω)
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Lecture 14 Slide #6
Input/Output Behavior of
LTI System in Frequency Domain
1
x[n] =
2π
∫
<2 π >
X(Ω)e
jΩn
1
y[n] =
2π
dΩ
H(Ω)
∫
H (Ω)X(Ω)e jΩn dΩ
<2 π >
1
y[n] =
2π
∫
Y (Ω)e jΩn dΩ
<2 π >
Y (Ω) = H (Ω)X(Ω)
Compare with y[n]=(h*x)[n]
Again, convolution in time
has mapped to
multiplication in frequency
6.02 Fall 2012
Lecture 14 Slide #7
Magnitude and Angle
Y (Ω) = H (Ω)X(Ω)
| Y (Ω) |= |H (Ω) |. | X(Ω) |
and
< Y (Ω) = < H (Ω)+ < X(Ω)
6.02 Fall 2012
Lecture 14 Slide #8
Core of the Story
1. A huge class of DT and CT signals
can be written --- using Fourier transforms --- as a
weighted sums of sinusoids (ranging from very slow to very fast)
or (equivalently, but more compactly) complex exponentials.
The sums can be discrete ∑ or continuous ∫ (or both).
2. LTI systems act very simply on sums of sinusoids:
superposition of responses to each sinusoid, with the
frequency response determining the frequency-dependent
scaling of magnitude, shifting in phase.
6.02 Fall 2012
Lecture 14 Slide #9
Loudspeaker Bandpass Frequency Response
SPL versus Frequency
(Speaker Sensitivity = 85dB)
100
97
94
SPL (dB)
91
88
85
82
79
-3dB @ 56.5Hz
76
-3dB @ 12.5k Hz
73
70
10
100
1,000
10,000
100,000
Frequency (Hz)
Image by MIT OpenCourseWare.
F ll 2012
6.02 Fall
Lecture 14 Slide #10
6.02 Fall 2012
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Commons license. For more information, see http://ocw.mit.edu/fairuse.
Lecture 14 Slide #11
Spectral Content of Various Sounds
Human Voice
Cymbal Crash
Snare Drum
Bass Drum
Guitar
Bass Guitar
Synthesizer
Piano
13.75 Hz- 27.5 Hz27.5 Hz
55 Hz
55 Hz110 Hz
110 Hz220 Hz
220 Hz440 Hz
440 Hz- 880 Hz- 1,760 Hz- 3,520 Hz- 7,040 Hz- 14,080 Hz880 Hz 1,760 Hz 3,520 Hz 7,040 Hz 14,080 Hz 28,160 Hz
Image by MIT OpenCourseWare.
6.02 Fall 2012
Lecture 14 Slide #12
Connection between CT and DT
The continuous-time (CT) signal
x(t) = cos( ωt) = cos(2πft)
sampled every T seconds, i.e., at a sampling
frequency of fs = 1/T, gives rise to the discrete-time
(DT) signal
x[n] = x(nT) = cos(ωnT) = cos(Ωn)
So
Ω = ωΤ
and Ω = π corresponds to ω = π/T or f = 1/(2T) = fs/2
6.02 Fall 2012
Lecture 14 Slide #13
Signal x[n] that has its frequency content
uniformly distributed in [–Ω
Ω c , Ωc]
1
x[n] =
2π
1
=
2π
∫
X(Ω)e
jΩn
dΩ
<2 π >
ΩC
∫
1⋅ e jΩn dΩ
−ΩC
sin(ΩC n)
=
, n≠0
πn
= ΩC / π
6.02 Fall 2012
, n=0
DT “sinc” function
(extends to ±∞ in time, falls off only as 1/n)
Lecture 14 Slide #14
x[n] and X(Ω
Ω)
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Lecture 14 Slide #15
X(Ω) and x[n]
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Lecture 14 Slide #16
Fast Fourier Transform (FFT) to compute
samples of the DTFT for
signals of finite duration
P−1
X( Ωk ) = ∑ x[m]e
m=0
− jΩk m
(P/2)−1
,
1
jΩk n
x[n] =
X(Ω
)e
∑
k
P k=−P/2
where Ωk = k(2π/P), P is some integer (preferably a power of 2)
such that P is longer than the time interval [0,L-1] over which
x[n] is nonzero, and k ranges from –P/2 to (P/2)–1 (for even P).
Computing these series involves O(P2) operations – when P gets
large, the computations get very s l o w….
Happily, in 1965 Cooley and Tukey published a fast method for
computing the Fourier transform (aka FFT, IFFT), rediscovering
a technique known to Gauss. This method takes O(P log P)
operations.
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P = 1024, P2 = 1,048,576, P logP ≈ 10,240
Lecture 14 Slide #17
Where do the Ωk live?
e.g., for P=6 (even)
Ω3
Ω2
Ω1
Ω2 Ω1
0
–
exp(jΩ2)
exp(jΩ3)
= exp(jΩ3)
–1
exp(jΩ2)
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Ω0
j
.
–j
Ω3
exp(jΩ1)
1
exp(jΩ0)
exp(jΩ1)
Lecture 14 Slide #18
Spectrum of Digital Transmissions
(scaled version of DTFT samples)
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Lecture 14 Slide #19
Spectrum of Digital Transmissions
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Lecture 14 Slide #20
Observations on previous figure
•
The waveform x[n] cannot vary faster than the step change every 7
samples, so we expect the highest frequency components in the
waveform to have a period around 14 samples. (The is rough and
qualitative, as x[n] is not sinusoidal.)
•
A period of 14 corresponds to a frequency of 2/14 = /7, which
is 1/7 of the way from 0 to the positive end of the frequency axis
at (so k approximately 100/7 or 14 in the figure). And that
indeed is the neighborhood of where the Fourier coefficients drop
off significantly in magnitude.
•
There are also lower-frequency components corresponding to the
fact that the 1 or 0 level may be held for several bit slots.
•
And there are higher-frequency components that result from the
transitions between voltage levels being sudden, not gradual.
6.02 Fall 2012
Lecture 14 Slide #21
Effect of Low-Pass Channel
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6
02 Fall 2012
Lecture 14 Slide #22
How Low Can We Go?
7 samples/bit 14 samples/period k=(N/14)=(196/14)=14
6.02 Fall 2012
Lecture 14 Slide #23
MIT OpenCourseWare
http://ocw.mit.edu
6.02 Introduction to EECS II: Digital Communication Systems
Fall 2012
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