G 3db bandwidth 4 noise bandwidth wn p s fc where

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Unformatted text preview: width Square-Root Raised-Cosine 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 Null-to-Null 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 Wi-Fi 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 Wi-Fi 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 time-bandwidth product for various definitions of bandwidth. For example, for the null-to-null bandwidth Wn we show Wn T . Pulse Shape Rectangular Sinusoidal Square-Root 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.

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