Frequency-Domain Analysis - Introduction to Digital Signal Processing and Filter Design - page 183

What is the magnitude at 6600 Hz? Compute the sample X 8 ( 2 ) of its
8-point DFT.
3.42 Given the 6-point DFT of f(n) , as given below compute the value of f( 3 ) :
F( 0 ) = 10 . 0 ; F( 1 ) =− 3 . 5 − j 2 . 6 ; F( 2 ) =− 2 . 5 − j 0 . 866

F( 3 ) =− 2 . 0 ; F( 4 ) =− 2 . 5 + j 0 . 866 ; F( 5 ) =− 3 . 5 + j 2 . 6

3.43 Compute the 6-point IDFT of X(k) given below:
X(k) ={ 3 + j 0 − 1 + j 0 − 0 + j 1 . 732 5 + j 00 − j 1 . 732

− 1 − j 0 }

If the N -point DFT of a real sequence x(n) is X N (k) , prove that the
DFT of x((

3.44

n)) N is X N (k) , using the property x(( − − n)) N = X(N − n) .

Show that the DFT of the even part x e (n) = [ x(n) + x( − n) ] / 2isgivenby

Re X(k) and the DFT of the odd part x o (n) = [ x(n) − x( − n) ] / 2isgiven

by j Im X(k) .
3.45 Find the even part and odd part of the following functions:
x 1 (n)

={ 1 − 1201 }

x 2 (n) ={ − − }

121 10 201

x 3 (n) ={ 11 − 13 }

}
3.46 Determine which of the following functions have real-valued DFT and
which have imaginary-valued DFT:
x 1 (n)

x 4 (n) ={ 012 − 110

={ 10 . 510010 . 5 }

x 2 (n) ={ 10 . 5 − 1101 − 1 . 5 }

x 3 (n) ={

00 . 5

− − −

0 . 5

}

110 11

}
3.47 Compute the 4-point DFT and 8-point DFT of x(n)

x 4 (n) ={ 1200100 − 2

={ 1 . 5 − 1 . 5 } .

Plot their magnitudes and compare their values.
3.48 Calculate the 5-point DFT of the x(n) ={ 1 . 5 − 1 . 5 } above.

3.49 Calculate the 6-point DFT of x(n) ={ 11 . 50 − 0 . 5 } .

3.50 Given the following samples of the 8-point DFT
X( 1 ) = 1 . 7071 − j 1 . 5858

X( 3 ) = 0 . 2929 + j 4 . 4142

j 2
find the values of X( 2 ), X( 5 ) ,and X( 7 ) .
3.51 Given the values of X( 4 ), X( 13 ), X( 17 ), X( 65 ), X( 81 ) ,and X( 90 ) of an
128-point DFT function, what are the values of X( 124 ), X( 63 ) , X( 115 ) ,
X( 38 ) , X( 111 ) ,and X( 47 ) ?

X( 6 ) =− 0 +