 Home
 Products
 Integration
 Tutorial
 Barcode FAQ
 Purchase
 Company
( b ) By the definition of h[nl [Eq. (2.3011 and Eq. (6.145) we obtain in VS .NET
( b ) By the definition of h[nl [Eq. (2.3011 and Eq. (6.145) we obtain Scan QR Code ISO/IEC18004 In VS .NET Using Barcode Control SDK for VS .NET Control to generate, create, read, scan barcode image in .NET applications. QR Drawer In VS .NET Using Barcode generation for .NET Control to generate, create QR Code 2d barcode image in .NET applications. h [ n ] = 6 [ n ] + 6 [ n  11 h [ n ]= Denso QR Bar Code Scanner In VS .NET Using Barcode recognizer for .NET Control to read, scan read, scan image in VS .NET applications. Barcode Generation In .NET Framework Using Barcode printer for Visual Studio .NET Control to generate, create barcode image in Visual Studio .NET applications. Osnsl otherwise
Recognize Bar Code In .NET Framework Using Barcode reader for Visual Studio .NET Control to read, scan read, scan image in .NET applications. Generating QR Code In Visual C# Using Barcode creator for VS .NET Control to generate, create QR Code ISO/IEC18004 image in .NET applications. From Eq. (6.146) Paint Denso QR Bar Code In Visual Studio .NET Using Barcode maker for ASP.NET Control to generate, create Denso QR Bar Code image in ASP.NET applications. Encode QR Code In Visual Basic .NET Using Barcode generator for .NET Control to generate, create QR Code 2d barcode image in .NET applications. xlnl
Creating Universal Product Code Version A In Visual Studio .NET Using Barcode encoder for .NET Control to generate, create UPCA image in .NET applications. USS Code 39 Generation In .NET Framework Using Barcode drawer for .NET Control to generate, create Code 39 Full ASCII image in VS .NET applications. Fig. 618 Matrix 2D Barcode Generation In VS .NET Using Barcode creator for Visual Studio .NET Control to generate, create Matrix Barcode image in .NET applications. USD3 Drawer In Visual Studio .NET Using Barcode generation for .NET framework Control to generate, create Code 93 Full ASCII image in .NET applications. CHAP. 61
Barcode Drawer In None Using Barcode generation for Software Control to generate, create bar code image in Software applications. Printing Data Matrix ECC200 In None Using Barcode drawer for Word Control to generate, create Data Matrix 2d barcode image in Office Word applications. FOURIER ANALYSIS OF DISCRETETIME SIGNALS AND SYSTEMS
Barcode Creator In Java Using Barcode creation for BIRT Control to generate, create bar code image in BIRT reports applications. Print UCC  12 In None Using Barcode printer for Excel Control to generate, create USS128 image in Office Excel applications. 9 ( ~ ) =  Barcode Generation In Java Using Barcode printer for Java Control to generate, create bar code image in Java applications. Making Code 128 Code Set C In Visual Studio .NET Using Barcode generator for ASP.NET Control to generate, create Code 128 Code Set C image in ASP.NET applications. IRI IT
Creating ECC200 In Java Using Barcode drawer for Java Control to generate, create Data Matrix image in Java applications. Creating EAN 128 In Java Using Barcode maker for Java Control to generate, create UCC.EAN  128 image in Java applications. which are sketched in Fig. 619. Fig. 619 Let R, , be the 3dB bandwidth of the system. Then by definition (Sec. 5.7) , we obtain
i13dB =
We see that the system is a discretetime wideband lowpass finite impulse response (FIR) filter (Sec. 2 . 9 0 . 6.35. Consider the discretetime LTI system shown in Fig. 620. where a is a constant and
Fig. 620 FOURIER ANALYSIS O F DISCRETETIME SIGNALS AND SYSTEMS [CHAP. 6
( a ) Find the frequency response H(S1) of the system. ( b ) Find the impulse response h [ n ]of the system. ( c ) Sketch the magnitude response ( H ( a ) ( the system for a of
( a ) From Fig. 620 we have y [ n ]  a y [ n  11 = x [ n ] = 0.9 and a
= 0.5. (6.147) Taking the Fourier transform of Eq. (6.147)and by Eq. (6.771,we have
( b ) Using Eq. (6.371, we obtain
h [ n ]= a n u [ n ] From Eq. (6.148) which is sketched in Fig. 621 for a = 0.9 and a = 0.5. We see that the system is a discretetime lowpass infinite impulse response (IIR) filter (Sec. 2 . 9 0 n 2
Fig. 621 6.36. Let h L p F [ nbe the impulse response of a discretetime 10~4pass ] filter with frequency . response H L p F ( R )Show that a discretetime filter whose impulse response h [ n ] is given by h[nl = ( l)"hLPF[nl
is a highpass filter with the frequency response
H ( S 1 )= H L P F ( a
CHAP. 61 FOURIER ANALYSIS O F DISCRETETIME SIGNALS AND SYSTEMS
Since  1 = el", we can write
( 6.152) h [ n ] = (  l ) " h L P F [ n ] eJ""hLPF[n] = Taking the Fourier transform of Eq. (6.152) and using the frequencyshifting property (6.44), we obtain H(R)= H L P F ( R  ~ ) which represents the frequency response of a highpass filter. This is illustrated in Fig. 622. + a, nR, Fig. 622 Transformation of a lowpass filter to a highpass filter.
6.37. Show that if a discretetime lowpass filter is described by the difference equation
then the discretetime filter described by
is a highpass filter.
Taking the Fourier transform of Eq. (6.153),we obtain the frequency response H L p F ( Rof ) the lowpass filter as If we replace R by ( R  a ) in Eq. (6.155), then we have
which corresponds to the difference equation
FOURIER ANALYSIS O F DISCRETETIME SIGNALS AND SYSTEMS [CHAP. 6
6.38. Convert the discretetime lowpass filter shown in Fig. 618 (Prob. 6.34) to a highpass filter. From Prob. 6.34 the discretetime lowpass filter shown in Fig. 618 is described by [Eq.
( 6.145 Using Eq. (6.154), the converted highpass filter is described by which leads to the circuit diagram in Fig. 623. Taking the Fourier transform of Eq. (6.157) and by Eq. (6.77), we have From Eq. (6.158) and which are sketched in Fig. 624. We see that the system is a discretetime highpass FIR filter.
Fig. 623 6.39. The system function H ( z ) of a causal discretetime LTI system is given by
where a is real and la1 < 1. Find the value of b so that the frequency response H ( R ) of the system satisfies the condition IH(n)l= 1 Such a system is called an allpass filter. By Eq. (6.34) the frequency response of the system is
all R
(6.160) CHAP. 6 FOURIER ANALYSIS OF DISCRETETIME SIGNALS AND SYSTEMS 1
Fig. 624 Then, by E . (6.160) q
which leads to
J b+ e]'I= 11  aejnl
Ib+cosnjsinRI=Il acosR+jasinRl
1 + b 2 + 2bcosR= 1 + a 2  2acosO
(6.162) and we see that if b = a, Eq. (6.162)holds for all R and Eq. (6.160)is satisfied.
6.40. Let h [ n ] be the impulse response of an FIR filter so that
h [ n ]=0 n<O,nrN
Assume that h [ n ] is real and let the frequency response H ( R ) be expressed as
H ( I 2 ) = 1H ( f l ) ) e ~ ~ ( ~ ) ( a ) Find the phase response 8 ( R ) when h [ n ] satisfies the condition [Fig. 625(a)] h[n]= h [ N  1  n ] (6.163) ( b ) Find the phase response B ( R ) when h [ n ] satisfies the condition [Fig. 625(b)] h [ n ]=  h [ N  1  n ] (6.164) FOURIER ANALYSIS OF DISCRETETIME SIGNALS AND SYSTEMS [CHAP. 6
N even
N I
I I I I I I
N odd
Fig. 625 Taking the Fourier transform of Eq. (6.163) and using Eqs. (6.431, (6.461, and (6.62).we obtain
H ( R ) = H * ( R ) ej(N')R
or Thus, IH(f))le1flfl)= ) H ( n ) ( e  i o ( ~ ~ e  ~ ( N  I ) n e(n)= e(n)  ( N i ) n
e ( n )= +(N 1 ) ~
which indicates that the phase response is linear. ( b ) Similarly, taking the Fourier transform of Eq. (6.164, we get ~ ( n= ) H * ( R ) e  ~ ( "  ' ) f l or Thus, I H ( f l ) l e i 0 ( n ) , IH(n)(e~nel@(fl)e~(Nl)fl
e(n)= T  q n )  ( N  i p
which indicates that the phase response is also linear.
CHAP. 61 FOURIER ANALYSIS OF DISCRETETIME SIGNALS AND SYSTEMS
6.41. Consider a threepoint movingaverage discretetime filter described by the difference
equation
Find and sketch the impulse response h [ n ]of the filter. ( b ) Find the frequency response H(IR) of the filter. (c) Sketch the magnitude response IH(IR)I and the phase response 8(IR) of the filter.

