Signals and Systems - Electrical Engineering

354 CHAPTER 5: Frequency Analysis: The Fourier Transform

which is imaginary function of, thus its computational importance. Show thatY()is also odd as a

function of.

(e) FindY()from the above equation (called the sine transform). Verify that your results are correct by

finding the Fourier transform ofz(t)=x(t)+y(t)directly and using the above results.

(f) What advantages do you see to the cosine and sine transforms? How would you use the cosine and the

sine transforms to compute the Fourier transform of any signal, not necessarily even or odd? Explain.

5.8. Time versus frequency—MATLAB

The supports in time and in frequency of a signalx(t)and its Fourier transformX()are inversely

proportional. Consider a pulse

x(t)=

1

T 0

[u(t)−u(t−T 0 )]

(a) LetT 0 = 1 andT 0 = 10 and find and compare the corresponding|X()|.

(b)Use MATLAB to simulate the changes in the magnitude spectrum when T 0 = 10 k for

k=0,..., 4forx(t). ComputeX()and plot its magnitude spectra for the increasing values of

T 0 on the same plot. Explain the results.

5.9. Smoothness and frequency content—MATLAB

The smoothness of the signal determines the frequency content of its spectrum. Consider the signals

x(t)=u(t+0.5)−u(t−0.5)

y(t)=( 1 +cos(πt))[u(t+0.5)−u(t−0.5)]

(a) Plot these signals. Can you tell which one is smoother?

(b)FindX()and carefully plot its magnitude|X()|versus frequency.

(c)FindY()(use the Fourier transform properties) and carefully plot its magnitude|Y()|versus

frequency.

(d)Which one of these two signals has higher frequencies? Can you now tell which of the signals is

smoother? Use MATLAB to decide. Makex(t)andy(t)have unit energy. Plot20 log 10 |Y()|and

20 log 10 |X()|using MATLAB and see which of the spectra shows lower frequencies.

5.10. Smoothness and frequency—MATLAB

Let the signalsx(t)=r(t+ 1 )− 2 r(t)+r(t− 1 )andy(t)=dx(t)/dt.

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

(b)FindX()and carefully plot its magnitude spectrum. IsX()real? Explain.

(c)FindY()(use properties of Fourier transform) and carefully plot its magnitude spectrum. IsY()real?

Explain.

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

20 log 10 |Y()|and20 log 10 |X()|and decide. Would you say in general that computing the derivative

of a signal generates high frequencies or possible discontinuities?

5.11. Integration and smoothing—MATLAB

Consider the signal

x(t)=u(t+ 1 )− 2 u(t)+u(t− 1 )

and let

y(t)=

∫t

−∞

x(τ)dτ