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barcode generator for ssrs Therefore, the energy in x ( n ) is in Software
Therefore, the energy in x ( n ) is Code128 Decoder In None Using Barcode Control SDK for Software Control to generate, create, read, scan barcode image in Software applications. Code 128B Drawer In None Using Barcode generator for Software Control to generate, create Code 128B image in Software applications. A complex bandpass analog signal x a ( t )has a Fourier transform that is nonzero over the frequency range [ Q l , Q2] as shown in the figure below. Scan Code128 In None Using Barcode recognizer for Software Control to read, scan read, scan image in Software applications. ANSI/AIM Code 128 Printer In Visual C#.NET Using Barcode maker for Visual Studio .NET Control to generate, create Code128 image in Visual Studio .NET applications. The signal is sampled to produce the sequence x ( n ) = xa(nT,). Create Code 128 Code Set B In .NET Using Barcode creator for ASP.NET Control to generate, create USS Code 128 image in ASP.NET applications. Code 128C Encoder In .NET Using Barcode creation for VS .NET Control to generate, create USS Code 128 image in .NET framework applications. ( a ) What is the smallest sampling frequency that can be used so that x a ( t ) may be recovered from its samples x(n) Code128 Creator In VB.NET Using Barcode creator for .NET Control to generate, create ANSI/AIM Code 128 image in VS .NET applications. Code128 Generation In None Using Barcode generator for Software Control to generate, create Code 128A image in Software applications. SAMPLING
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as illustrated below.
Q2Qs
In order for there to be no interference between the shifted spectra, it is necessary that
If this condition is satisfied, xa(t) may be uniquely reconstructed from xs(t) using a bandpass filter with a frequency response as shown below. (b) With a sampling frequency Q, = Q2  Q l , the reconstruction filter is a complex bandpass filter with an impulse response ha(!) = Ts s i n ( Q s t / 2 ) e  j ( ~ 2 + ~ ~ ~
CHAP. 31
SAMPLING Therefore, the output of the reconstruction filter, which produces the complex bandpass signal x,(t), is Given a realvalued bandpass signal x , ( t ) with X a ( f ) = 0 for If l < f i and If 1 > f2, the Nyquist sampling theorem says that the minimum sampling frequency is fs = 2 f2. However, in some cases, the signal may be sampled at a lower rate. (a) Suppose that fl = 8 kHz and f2 = 10 kHz. Make a sketch of the discretetime Fourier transform of x ( n ) = x,(nTs) if fs = I /T, = 4 kHz. ( 6 ) Define the bandwidth of the bandpass signal to be and the center frequency to be
Show that if f , > B / 2 and f2 is an integer multiple of the bandwidth B , no aliasing will occur if x a ( t ) is sampled at a sampling frequency fs = 2 8 . (c) Repeat part (b) for the case in which f2 is not an integer multiple of the bandwidth B .
(a) Let x u ( [ )have a spectrum as shown in the figure below.
The spectrum of the sampled signal
which is formed by shifting X,(f ) by integer multiples of the sampling frequency and summing. With f, = 4 kHz, we have the spectrum sketched below. Note that X,( f) is not aliased. Therefore, with the appropriate processing of x,(r), the signal x,(r) may be recovered from its samples. Finally, the DTFT of the discretetime sequence x ( n ) = x,(nTs) is SAMPLING
which is sketched below.
[CHAP. 3
x(c'") (b) If fi is an integer multiple of B , we may express f i and f2 as follows: With a sampling frequency of f, = 2 B , the sampled signal has a spectrum
Because X a ( f ) is nonzero only for (I  l ) B < If 1 < l B , there is only one term in the sum that contributes to X,( f ) in the frequency range O< f <B and only one term that contributes to the frequency range B if < 0 (draw a picture as in part (a) to see this clearly). Therefore, there is no aliasing, and x,(t) may be sampled without aliasing if a sampling frequency f, = 2B. (c) If f2 is not an integer multiple of B, we may always increase B until this is the case. Specifically, let where
1.1 is defined to be the "integer part." Now, if we simply increase B to B' where
we have the case described in part ( b ) where fi is an integer multiple of the bandwidth. Thus, x,(t) may be sampled without aliasing a sampling frequency of Determine the minimum sampling frequency for each of the following bandpass signals:

