In other words find the optimum value of c in the approximation ft≈cxt so that the error signal energy is minimum.. Ch-3: Fourier series representation of periodic signals P3.3.. If xt a
Trang 1Signal & Systems - FEEE, HCMUT – Semester: 02/10-11
P3.1 For the signals f(t) and x(t) depicted in Figure P3.1, find the
component of the form x(t) contained in f(t) In other words find
the optimum value of c in the approximation f(t)≈cx(t) so that the
error signal energy is minimum Find the error signal e(t) and it
energy Ee Show that the error signal is orthogonal to x(t), and that
Ef=c2Ex+Ee Can you explain this result in terms of vector?
P3.2 Repeat P3.1 if x(t) is sinusoid pulse shown in Figure P3.2.
Ch-3: Fourier series representation of periodic signals
P3.3 If x(t) and y(t) are orthogonal, then show that the energy of
the signal x(t)+y(t) is identical to the energy of the signal x(t)-y(t)
and is given by Ex+Ey Explain this result using vector concepts In
general, show that for orthogonal signal x(t) and y(t) and for any
pair of arbitrary constant c1and c2, the energies of c1x(t)+c2y(t) and
c1x(t)-c2y(t) are identical, given by:
c E +c E
Trang 2Signal & Systems - FEEE, HCMUT – Semester: 02/10-11
P3.4 Figure P3.4(a) shows the first eight functions in Walsh function
set Represent f(t) in Figure P3.4(b) over interval [0,1] using a Walsh
Fourier series using 8 basis functions Compute the energy of e(t), the
error in the approximation using the first N non-zero terms in the series
for N=1, 2, 3 and 4
Signal & Systems - FEEE, HCMUT – Semester: 02/10-11
P3.5 Determine the cross-correlation of the following signals:
Ch-3: Fourier series representation of periodic signals
x(t)=e u(t), and y(t)=e u(t);a>0
P3.6 Determine the cross-correlation of the following signals:
-at t
2T x(t)=rect( ), and y(t)=e u(t);a>0
P3.7 Consider the signal f(t)=rect(t-1/2) Determine its
autocorrelation function and its energy using this function
P3.8 Find the autocorrelation of the signal f(t)=cos(ω0t)rect(t/T)
Trang 3Signal & Systems - FEEE, HCMUT – Semester: 02/10-11
P3.9 A continuous-time periodic signal f(t) is real valued and has a
fundamental period T=8 The nonzero Fourier series coefficients fo
f(t) are: D1=D-1=2, D3=(D-3)*=j4
Express f(t) in the form:
n 0
f(t)= C cos( k n)
n
t
+∞
=
+
∑
P3.10 Using the Fourier series analysis to calculate the coefficients
Dnfor the continuous-time periodic signal
1.5; 0 t<1 f(t)=
-1.5; 1 t<2
≤
≤
with fundamential frequency ω0=π
P3.11 Determine the Fourier series representation for each of the
periodic signals depicted in Figure P3.11
Ch-3: Fourier series representation of periodic signals
Trang 4Signal & Systems - FEEE, HCMUT – Semester: 02/10-11
P3.12 In each of the following, we specify the Fourier series
coefficients of a continuous-time signal that is periodic with period
4 Determine the signal f(t) in each case
n n sin(n /4)
n
0; n=0
(a) D =
(j) ππ ; otherwise
sin(n /8) n
n
jn; |n|<3
(c) D =
0; otherwise
n
1; n even
(d) D =
2; n odd
Signal & Systems - FEEE, HCMUT – Semester: 02/10-11
P3.13 Consider a continuous-time LTI system whose frequency
response is
Ch-3: Fourier series representation of periodic signals
-sin(4 )
ω
∞
∫
If the input to the system is a periodic signal f(t)= 1; 0 t<4
-1; 4 t<8
≤
≤
with period T=8, determine the corresponding system output y(t)
P3.14 Consider a continuous-time ideal low-pass filter whose
frequency rsponse is
1; | | 100 H(j )=
0; | |>100
ω ω
ω
≤
When the input to this filter is a signal f(t) with fundamental period
T=π/6 and Fourier series coefficients Dn, it is found that
f(t)→y(t)=f(t)
For what values of n is it guaranteed that Dn=0?
Trang 5Signal & Systems - FEEE, HCMUT – Semester: 02/10-11
P3.15 Consider an LTI system with impulse response h(t)=e-4tu(t)
Find the Fourier series representation of the output y(t) for each of
the following inputs:
(a) f(t)=cos(2 t)π
(b) f(t)=sin(4 t)+cos(6 t+ /4)π π π
n=
(c) f(t)= δ(t n)
+∞
−∞
−
∑
n n=
(d) f(t)= ( 1) δ(t n)
+∞
−∞
∑
(e) f(t) is periodic square wave dipicted in Figure P3.15
f(t)
Figure P3.15