Signals and Systems - Electrical Engineering

634 C H A P T E R 10: Fourier Analysis of Discrete-Time Signals and Systems

10.14. Sinusoidal form of DTFT

A triangular pulse is given by

t[n]=







3 +n − 2 ≤n≤− 1

3 −n 0 ≤n≤ 2

0 otherwise

(a) The pulse can be written as

t[n]=

∑∞

k=−∞

Akδ[n−k]

Find the{Ak}coefficients.

(b) Find a sinusoidal expression for the DTFT oft[n]—that is,

T(ejω)=B 0 +

∑∞

k= 1

Bkcos(kω)

Express the coefficientsB 0 andBkin terms of theAkcoefficients.

10.15. DTFT and Z-transform—MATLAB

Letx[n]=r[n]−r[n−3]−u[n−3]wherer[n]is the ramp signal.

(a) Carefully plotx[n]and find its Z-transformX(z).

(b) Ify[n]=x[−n], giveY(z)in terms ofX(z).

(c) Use the above results to find the DTFT ofx[n],x[−n], andx[n]+x[−n]. Find the magnitude of each

of these DTFTs and then use MATLAB to compute them and plot them.

10.16. Computations from DTFT definition

For simple signals it is possible to obtain some information on their DTFTs without computing them. Let

x[n]=δ[n]+ 2 δ[n−1]+ 3 δ[n−2]+ 2 δ[n−3]+δ[n−4]

(a) FindX(ej^0 )andX(ejπ)without computing the DTFTX(ejω).

(b) Find

∫π

−π

|X(ejω)|^2 dω

(c) Find the phase ofX(ejω). Is it linear?

10.17. DTFT of even and odd functions

A signal

x[n]=0.5nu[n]

is neither even nor odd.

(a) Find the evenxe[n]and the oddxo[n]components ofx[n], and carefully plot them.

(b) Find the Z-transforms ofxe[n]andxo[n], and from them find the DFTsXe(ejω)andXo(ejω). Are they

real or imaginary?

(c) Sincex[n]=xe[n]+xo[n]so thatX(ejω)=Xe(ejω)+Xo(ejω), how dos the real and the imaginary

parts ofX(ejω)relate toXe(ejω)andXo(ejω)? Explain.

(d) Use Parseval’s result to obtain thatEx=Exe+Exoi.e., the energy of the signal is the sum of the

energies of its even and odd components.