Discrete-time Fourier series and transform - Continuous and Discrete Time Signals and Systems - page 520

The magnitude spectrum of an ideal lowpass filter is unity within its pass band
( Ω

≤ π ).
The converse of the ideal lowpass filter is the ideal highpass filter, which
blocks all frequency components below a certain cut-off frequency Ω > Ω c in
the applied input sequence. All frequency components Ω

≤

Ω c ) and zero within its stop band ( Ω c < Ω

Ω c are left unatten-
uated in the output response of an ideal highpass filter. The magnitude spectrum
of an ideal highpass filter is unity within the pass band ( Ω c

≥

≤ ≤ π ) and zero

Ω

within the stop band (0 ≤

Ω < Ω c ).
Section 11.8 compared the Fourier representations of CT and DT periodic
and aperiodic waveforms. We showed that the Fourier representations of peri-
odic waveforms are discrete, whereas the Fourier representations of discrete
waveforms are periodic.

Problems

11.1 Determine the DTFS representation for each of the following DT peri-
odic sequences. In each case, plot the magnitude and phase of the DTFS
coefficients.
(i) x [ k ] = k

for 0 ≤ k ≤ 5 and x [ k + 6] = x [ k ];

1( 0 ≤ k

≤ 2)

(ii) x [ k ] = 0 . 5(3 ≤ k ≤ 5) and x [ k + 9] = x [ k ];

0( 6 ≤ k ≤ 8)

2 π
7

k + π
4

(iii) x [ k ] = 3 sin ;

(iv) x [ k ] = 2e j ( 5 3 k + 4 ) ;

m =−∞ δ ( k − 5 m );
(vi) x [ k ] = cos(10 π k / 3) cos(2 π k / 5);
(vii) x [ k ] = cos(2 π k / 3) .

∞

(v) x [ k ] =

11.2 Given the following DTFS coefficients, determine the DT periodic
sequence in the time domain:

1( 0 ≤ k

≤ 2)

(i) D n = 0 . 5(3 ≤ k ≤ 5) and D n + 9 = D n ;

0( 6 ≤ k ≤ 8)

1 − j0 . 5 n =− 1)

( n = 0)

(ii) D n = and D n + 7 = D n ;

1 + j0 . 5 n = 1)

( 2 ≤ n ≤ 5)

= 1 + 3

π n
8

(iii) D n

4 sin

(0 ≤ n ≤ 6) and D n + 7 = D n ;

(iv) D n

= ( − 1) n D n ; (0 ≤ n ≤ 7) and D n + 8 =

(v) D n = e j n π/ 4

(0 ≤ n

≤ 7) =

D n .

and D n + 8