Signals and Systems - Electrical Engineering

Problems 355

(a) Plotx(t)andy(t).

(b)FindX()and carefully plot its magnitude spectrum. IsX()real? Explain. (Use MATLAB to do the

plotting.)

(c)FindY()and carefully plot its magnitude spectrum. IsY()real? Explain. (Use MATLAB to do the

plotting.)

(d)Determine from the above spectra which of these two signals is smoother. Use MATLAB to decide.

Would you say that in general by integrating a signal you get rid of higher frequencies, or smooth out

a signal?

5.12. Duality of Fourier transforms
As indicated by the derivative property, if we multiply a Fourier transform by(j)N, it corresponds to
computing anNthderivative of its time signal. Consider the dual of this property—that is, if we compute
the derivative ofX(), what would happen to the signal in the time?
(a) Letx(t)=δ(t− 1 )+δ(t+ 1 ). Find its Fourier transform (using properties)X().
(b)ComputedX()/dand determine its inverse Fourier transform.

5.13. Periodic functions in frequency
The duality property provides interesting results. Consider the signal

x(t)=δ(t+T 1 )+δ(t−T 1 )

(a) FindX()=F[x(t)]and plot bothx(t)andX().

(b)Suppose you then generate a signal

y(t)=δ(t)+

∑∞

k= 1

[δ(t+kT 0 )+δ(t−kT 0 )]

Find its Fourier transformY()and plot bothy(t)andY().

(c)Arey(t)and the corresponding Fourier transformY()periodic in time and in frequency? If so,

determine their periods.

5.14. Sampling signal
The sampling signal

δTs(t)=

∑∞

n=−∞

δ(t−nTs)

will be important in the sampling theory later on.

(a) As a periodic signal of periodTs, expressδTs(t)by its Fourier series.

(b)Determine then the Fourier transform1()=F[δTs(t)].

(c)PlotδTs(t)and1()and comment on the periodicity of these two functions.

5.15. Piecewise linear signals
The derivative property can be used to simplify the computation of some Fourier transforms. Let

x(t)=r(t)− 2 r(t− 1 )+r(t− 2 )

(a) Find and plot the second derivative with respect totofx(t), ory(t)=d^2 x(t)/dt^2.

(b)FindX()fromY()using the derivative property.

(c)Verify the above result by computing the Fourier transformX()directly fromx(t)using the Laplace

transform.