Trabalho LISTA DSP G2

Prévia do material em texto

π 
{ } 
 
Alunos: Rafael Cristiano Bartz Matrícula: 200810941 
Prof: Artur Cardoso Severo Disciplina: Processamento Digital de Sinais 
 
Lista 2 
 
 
a) 
 
X [ N ]= 1 ∫ j d x (eJw ) eJwN dw 
2 π − π dw 
 
{u=eJwN e dw=JN eJwN } 
v=Jx (eJw )e dv= d x (eJw )dw 
dw 
 
X [ N ]= 1 
2 π 
π 
(ewN − Jx (eJw ))−∫ (Jx (eJw )eJwN JNdw) 
− π 
 
 
X [ N ]= 1 (eJ π N J x (eJN ) −e−J π N − Jx (e− J π )); eJ π =−1, logo : 
2 π 
 N ∫ 
X [ N ]=
 1 
 − J x (−1) −
 1 
Jx
 1 
 =
 1 
(−J x (−1)+1 Jx (−1))=0 
2 π ( 
 
e
JxN ( eN )) 2 π 
 
Na 2ª equação temos: 
 
X [ N ]= − 1 
π 
∫ Jx (eJw )eJwN JNdw ; Sabendo que JXJ =−1, logo : 
2 π − π 
 
π 
X [ N ]= x (eJw) eJwN dw 
2 π − π 
 
Assim, a DTFT de J
 d 
X (eJw) é N . X [ N ] 
dw 
 
 
 
 
 
 
A propriedade de multiplicação por N discreto 
 
 
NX [ N ] ⇔ J d X (eJw) 
dw 
Question 3 
(Difficulty: * ) A discrete sequence x[n] has for DTFT X(e:iw ). 
The real and imaginary paris of X( e:iw ) are : 
 
 
-lf o " 
 
 
 
 
 
 
 
" 
 
 
 
 
 
B y visua l i nspecli on of lhe p l ol s , t i ck ali the true statements about l he sequence x[n] : 
D x[n] is Hermitian- 
symmelric 
x[n] = x*[- n]. 
■ x[n]is 0- mean, 
i.e. 
L neZx[n] = O. 
O x[n] is real 
valued. 
, mjx(,,i-'JI 
Question 4 
 
(Difficulty: * ) 
 
lntroduction 
(optional , may skip to lhe question directty) 
 
 
You 'Nant to transmit a sequence x(n} thro ugh the air , for this you are provided a digital to 
analog converter (DAC) connected to an antenna. ln this exercise, you can assume the DAC is a 
black box converting a real-valued sequence s(n) into a "smooth" electrical signal s(t) : 
 
 
 
 
.,/11) 
- -H 
o .o.e 
s/11/ 
 
 
Question 
The sequence x(nJ is a sampled signal. it is rea l-vatued and its frequency content is limited to 
[- ir/ 8, ir/8). 
We are allotted a frequency-band which dictates the follo'AAng constraint : 
 
Frequency 
Frequency support of s/11/ support of , /11} Frequency support of s/11/ 
 
 
·• -3, / 4 ·•/2 -r/8 O • / 8 • / 2 r 
 
and so, a mystery function H must be inserted to convert x[n) into a sequence s[n] satisfying 
the following requisites : 
 
• The support ofthe DTFT of s[n] mus t be limited to [- 31r/ 4, - ir/ 2]U [ir/ 2, 3ir/ 4] 
• The sequence s(n]must be re al-valued {x(n) is real-va lued) 
 
Whic h of the follo'Mng input/o utput relationships for H meet the two requisites (check all 
correct answers) : 
L s[n] = ,!2i"" · x[n l 
e 
s[n ) = IDTF T{X (e(iw h / 8 l ) } 
 
■ s(n l = sin( 2!"n) · x[nl 
 
■ s(n l = cos( 58" n) • x[nl 
π 
 
 
Para y[n] a peridiocidade é: P=
 2 π 
=3 
4 π 
6 
 
2 
Para x[n] a peridiocidade é: P= 
2 π 
=1 
 
 
2 10 4 5 3 8 7 1 6 9