Unformatted text preview: width SquareRoot RaisedCosine Pulses The smaller α the longer the time for the pulse to die out.
The smaller α the narrower the spectrum
When α → 0 the spectrum becomes ﬂat and concentrated over
the interval [− 21 , 21 ]. This is called the Nyquist pulse.
T
T EECS 455 (Univ. of Michigan) Fall 2012 September 6, 2012 23 / 130 Lecture Notes 5 Bandwidth Deﬁnitions of Bandwidth for Digital Signals
∆ 1 NulltoNull bandwidth = bandwidth of main lobe of power spectral
density 2 99% power bandwidth containtment = bandwidth such that 1/2%
of power lies above upper band limit and 1/2% lies below lower
band limit 3 x dB bandwidth = bandwidth such that spectrum is x dB below
spectrum at center of band (e.g. 3dB bandwidth) 4 Noise bandwidth = WN = P /S (fc ) where P is total power and
S (fc ) is value of spectrum at f = fc . 5 6 ∆ ∆ ∆ ∆ Gabor bandwidth = σ where σ 2 =
∆ R∞
(f −fc )2 S (f )df
−∞
R∞
S (f )df
−∞ Absolute bandwidth = WA = min{W : S (f ) = 0∀f  > U } EECS 455 (Univ. of Michigan) Fall 2012 September 6, 2012 24 / 130 Lecture Notes 5 Bandwidth Deﬁnitions of Bandwidth for Digital Signals
S (f )
S (fc ) (Area under curve = P)
WN fc EECS 455 (Univ. of Michigan) Fall 2012 f September 6, 2012 25 / 130 Lecture Notes 5 Bandwidth WiFi Spectral Mask 0 dB
10 dB
20 dB
28 dB
40 dB
40 30 EECS 455 (Univ. of Michigan) 20 10 9 Fall 2012 1120 30 40 f MHz September 6, 2012 26 / 130 Lecture Notes 5 Bandwidth WiFi Spectral Mask with Sine Pulses
0 T=0.15 −5 −10 −20 −25 y S (f) (dB) −15 −30 −35 −40 −45 −50
−60 −40 −20 0 20 40 60 f (MHz) EECS 455 (Univ. of Michigan) Fall 2012 September 6, 2012 27 / 130 Lecture Notes 5 Bandwidth Rectangular Pulse Example SY (f ) = T sin2 π fT
= T sinc2 (fT )
(π fT )2 sinc((f − fc )T = 0 at π (f − fc )T = hπ h = ±1, ±2, ±3, . . .
n
f = fc + T , n = ±1, ±2, . . .
∆ Null to null bandwidth = width of main lobe of spectral density.
2
For PSK null to null bandwidth = T
∆ Fractional Power containment Bandwidth = width of frequency
band which leaves 1/2% of signal power above upper band limit
and 1/2% of signal power below band limit
.
For PSK 99% energy bandwidth = 20T56
We would like to ﬁnd modulation schemes which decrease the
bandwidth while retaining acceptable performance
EECS 455 (Univ. of Michigan) Fall 2012 September 6, 2012 28 / 130 Lecture Notes 5 Bandwidth Rectangular Pulse Example
In the table below we show the values of the timebandwidth product
for various deﬁnitions of bandwidth. For example, for the nulltonull
bandwidth Wn we show Wn T . Pulse Shape
Rectangular
Sinusoidal
SquareRoot
Raised Cosine
(α = 0.5) Bandwidth
1
2.0
3.0
1.5 Deﬁnition
2
20.56
1.18
1.268 3 35dB
35.12
3.24 4
1.00
0.62 5
∞
(0.5) 1
2 3 3dB
0.88
0.59
1.0 These three modulation schemes all have same error probability.
MSK has minimum possible Gabor bandwidth over all modulation schemes
whose basic pulse is limited to 2T seconds.
All of these have inﬁnite absolute bandwidth.
EECS 455 (Univ. of Michigan) Fall 2012 September 6, 2012 29 / 130 Lecture Notes 5 Bandwidth Shannon’s theorem revisited Theorem
(Shannon) For a white Gaussian noise channel there exist signals with
1
absolute bandwidth W conveying R = T bits of information per second
with arbitrarily small error probability provided
R= P
ER
1
< W log2 (1 +
) = W log2 (1 + b )
T
WN0
N0 W where Eb = PT is the energy per data bit and P is the power of the
signal EECS 455 (Univ. of Michigan) Fall 2012 September 6, 2012 30 / 130 Lecture Notes 5 Bandwidth Shannon’s theorem revisited P= 1
T →∞ 2T T lim −T R < W log 1 + R /W < log 1 + s2 (t )dt R Eb
W N0 → P
as W → ∞
N0 ln 2 R Eb
W N0 2R /W − 1
R
→ ln 2 as
→0
R /W
W
ln(2) ⇒ (Eb /N0 )dB = 10 log10 (ln(2)) = −1.6dB ⇒ Eb /N0 >
Eb /N0 = EECS 455 (Univ. of Michigan) Fall 2012 September 6, 2012 31 / 130 Lecture Notes 5 Up and Down Conversion Up and Down Conversion In communication systems typically the signals are generated at
baseband and then up converted to the desired carrier frequency. At
the receiver this process is reversed. EECS 455 (Univ. of Michigan) Fall 2012 September 6, 2012 32 / 130 Lecture Notes 5 Up and Down Conversion Up and Down Conversion xc (t ) × √ EECS 455 (Univ. of Michigan) yc (t ) 2 cos(2π fc t ) Fall 2012 September 6, 2012 33 / 130 Lecture Notes 5 Up and Down Conversion Up and Down Conversion √
yc (t ) = xc (t ) 2 cos(2π fc t )
√ 2
[Xc (f − fc ) + Xc (f + fc )]
2
√
Thus multiplication in the time domain by 2 cos(2π fc t ) shifts the
spectrum up and down by fc and reduces each part by 1/2.
Yc (f ) = EECS 455 (Univ. of Michigan) Fall 2012 September 6, 2012 34 / 130 Lecture Notes...
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This note was uploaded on 02/12/2014 for the course EECS 455 taught by Professor Stark during the Fall '08 term at University of Michigan.
 Fall '08
 Stark

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