*Ch3 Digital Processing of Continuous-Time Signals
*3.1 Show that if the spectrum Ga ( j) of g a (t ) (band-limited to  m ) also contained an
impulse at  m , the sampling rate T must be greater than 2 m to recover fully g a (t )
from the sampled version.
*3.2 The Nyquist frequency of continuous-time signal g a (t ) is  m . Determine the Nyquist
frequency of each of the following continuous-time signals derived from g a (t ) :
a)
y1 (t )  g a (t ) g a (t ) ;
b)
y2 (t )  g a (t / 3)
c)
y3 (t )   g a (t   ) g a ( )d
d)
y 4 (t ) 


dg a (t )
dt
*3.3 A finite-energy continuous-time signal g a (t ) is sampled at a rate satisfying the Nyquist
condition generating a discrete-time sequence g[n] . Develop the relation between the total
energy  g a (t ) of the continuous-time signal g a (t ) and the total energy  g [n ] of the
continuous-time signal g[n] .
*3.4 A continuous-time signal xa (t ) is composed of a linear combination of sinusoidal signal of
frequencies 300Hz, 500Hz, 1.2 kHz, 2.15 kHz and 3.5 kHz. The signal xa (t ) is sampled at
a 3-kHz rate, and the sampled sequence is passed through an ideal lowpass filter with a cutoff
frequency of 900Hz, generating a continuous-time signal ya (t ) . What are the frequency
components present in the reconstructed signal ya (t ) ?
*3.5 A continuous-time signal xa (t ) is composed of a linear combination of sinusoidal signal of
frequencies F1Hz, F2Hz, F3Hz and F4Hz. The signal xa (t ) is sampled at an 8-kHz rate, and
the sampled sequence is passed through an ideal lowpass filter with a cutoff frequency of 3.5
kHz, generating a continuous-time signal ya (t ) composed of three sinusoidal signals of
frequencies 150Hz, 400Hz and 925Hz, respectively. What are the possible values of F1, F2,
F3 and F4? Is your answer unique? If not, indicate another set of possible values of these
frequencies.
xa (t )  3sin( 32t )  2 cos( 24t )  4 sin( 120t )  8 cos(144t ) is
*3.6 The continuous-time signal
sampled at a 40 Hz rate, generating the sequence x[n ] . Determine the exact expression of
x[n ] .
*3.7 Consider the system of Fig. 1, where the input continuous-time signal xa (t ) has a
band-limited spectrum X a ( j) , as sketched in Fig.2 (a) and is being sampled at Nyquist rate.
j
The discrete-time processor is an ideal lowpass filter with a frequency response H (e ) , as
shown in Fig.2(b), and has a cutoff frequency
c   mT / 2 , where T is the sampling
period. Sketch as accurately as possible the spectrum Ya ( j) of the output continuous-time
signal ya (t ) .
x[n]
xa(t)
Ideal
sampler
DiscreteTime
Processor
y[n]
ya(t)
Ideal
interpolator
Fig. 1
H(e jω)
Xa(jΩ)
1
1
0.5
0.75Ωm
-Ωm
0
0.5Ωm
Ωm
ω
Ω
-ω m
0
(b)
(a)
Fig. 2
ωm