Introduction to signals - Continuous and Discrete Time Signals and Systems - page 53

Problems

1.1 For each of the following representations:
(i) z [ m , n , k ] ,
(ii) I ( x , y , z , t ) ,
establish if the signal is a CT or a DT signal. Specify the independent
and dependent variables. Think of an information signal from a physical
process that follows the mathematical representation given in (i). Repeat
for the representation in (ii).

1.2 Sketch each of the following CT signals as a function of the independent
variable t over the specified range:
(i) x 1( t ) = cos(3 π t / 4 + π/ 8) for − 1 ≤ t ≤ 2;

(ii) x 2( t ) = sin( − 3 π t / 8 + π/ 2) for − 1 ≤ t ≤ 2;

(iii) x 3( t ) = 5 t + 3 exp( − t ) for − 2 ≤ t ≤ 2;

(iv) x 4( t ) = (sin(3 π t / 4 + π/ 8)) 2

for − 1 ≤ t ≤ 2;

(v) x 5( t ) = cos(3 π t / 4) + sin( π t / 2) for − 2 ≤ t ≤ 3;

(vi) x 6( t ) = t exp( − 2 t ) for − 2 ≤ t ≤ 3 .

1.3 Sketch the following DT signals as a function of the independent variable
k over the specified range:
(i) x 1[ k ] = cos(3 π k / 4 + π/ 8) for − 5 ≤ k ≤ 5;

(ii) x 2[ k ] = sin( − 3 π k / 8 + π/ 2) for − 10 ≤ k ≤ 10;

− k

(iii) x 3[ k ] = 5 k + 3 for − 5 ≤ k ≤ 5;

(iv) x 4[ k ] = sin(3 π k / 4 + π/ 8) for − 6 ≤ k ≤ 10;

(v) x 5[ k ] = cos(3 π k / 4) + sin( π k / 2) for − 10 ≤ k ≤ 10;

− k

(vi) x 6[ k ] = k 4 for − 10 ≤ k ≤ 10 .

1.4 Prove Proposition 1.2.

1.5 Determine if the following CT signals are periodic. If yes, calculate the
fundamental period T 0 for the CT signals:
(i) x 1( t ) = sin( − 5 π t / 8 + π/ 2);
(ii) x 2( t ) = sin( − 5 π t / 8 + π/ 2) ;
(iii) x 3( t ) = sin(6 π t / 7) + 2 cos(3 t / 5);
(iv) x 4( t ) = exp( j(5 t + π/ 4));
(v) x 5( t ) = exp( j3 π t / 8) + exp( π t / 86);
(vi) x 6( t ) = 2 cos(4 π t / 5)

sin 2 (16 t / 3);
(vii) x 7( t ) = 1 + sin 20 t + cos(30 t + π/ 3) .

∗

1.6 Determine if the following DT signals are periodic. If yes, calculate the
fundamental period N 0 for the DT signals:
(i) x 1[ k ] = 5 ( − 1) k ;
(ii) x 2[ k ] = exp( j(7 π k / 4)) + exp( j(3 k / 4));
(iii) x 3[ k ] = exp( j(7 π k / 4)) + exp(j(3 π k / 4));
(iv) x 4[ k ] = sin(3 π k / 8) + cos(63 π k / 64);