Introduction to signals - Continuous and Discrete Time Signals and Systems - page 56

x 1 ( t ) x 2 ( t )

2.5

5

t t t t

−4 −3 −2 0 123 −4 −3 −2 0 123

−1 −1

(a) (b)

−2.5

x 4 ( t )

x 3 ( t )= e −1.5 t u ( t )

2.5

2

t t

t t

−12 −9 −6 −3 0 369 2

−4 −3 −2 −1 0 1234

−2.5

(d)

(c)

Fig. P1.17. Waveforms for
Problem 1.17.

(iii) x 3( t ) = rect( t / 6) + rect( t / 4) + rect( t / 2);
(iv) x 4( t ) = r ( t ) − r ( t − 2) − 2 u ( t − 4);
(v) x 5( t ) = (exp( − t ) − exp( − 3 t )) u ( t );
(vi) x 6( t ) = 3 sgn( t ) rect( t / 4) + 2 δ ( t + 1) − 3 δ ( t − 3).

1.19 (a) Sketch the following functions with respect to the time variable (if
a function is complex, sketch the real and imaginary components sep-
arately). (b) Locate the frequencies of the functions in the 2D complex
plane.
(i) x 1( t ) = e j2 π t + 3 ;
(ii) x 2( t ) = e j2 π t + 3 t ;
(iii) x 3( t ) = e

− j2 π t + j3 t ;
(iv) x 4( t ) = cos(2 π t + 3);
(v) x 5( t ) = cos(2 π t + 3) + sin(3 π t + 2);
(vi) x 6( t ) = 2 + 4 cos(2 π t + 3) − 7 sin(5 π t + 2).

1.20 Sketch the following DT signals:
(i) x 1[ k ] = u [ k ] + u [ k − 3] − u [ k − 5] − u [ k − 7];

∞

(ii) x 2[ k ] = δ [ k − m ];

m = 0

(iii) x 3[ k ] = (3 k − 2 k ) u [ k ];
(iv) x 4[ k ] = u [cos( π k / 8)];
(v) x 5[ k ] = ku [ k ];
(vi) x 6[ k ] = k ( u [ k + 4] − u [ k − 4]).

1.21 Evaluate the following expressions:

5 + 2 t + t 2
7 + t 2 + t 4 δ ( t − 1);

(i)

sin( t )
2 t

(ii) δ ( t );

ω 3 − 1
ω 2 + 2 δ ( ω − 5) .

(iii)