Selected Topics in Applied Signal Processing - Digital Signal Processing: An Introduction with MATLAB and Applications - page 139

| H ( e j2 π f ) |

1

f , Hz

0

− f s / 2 = −500 f s / 2 = 500

(a)

x ( t ) = sin(

ω o t )

t

0 0.2

x ( t ) = cos( ω o t )

(b)

Fig. 3.6 FIR implementation of the digital HT. a The HT magnitude response. b HT applied to
the sinusoid x ð t Þ¼ cos ð x o t Þ; which results in a 90 phase shift. Note that the first few samples of
x ð t Þ are incorrect, hence ignored in the plot

h=hd.*w_han;
num=h;
den=[1 zeros(1,M-1)];
H=freqz(num,den,f,fs);
Figure 3.6 shows the results of this implementation and its application to a
sinusoid.

3.3.2 The Analytic Signal

Most practical signals are real and have a positive frequency spectrum as well as a
mirror image negative frequency spectrum. It is possible to synthesize a so-called
analytic signal which has a no negative frequency content. The analytic signal
z(t) associated with a real signal x(t) is defined as:

z ð t Þ¼ x ð t Þþ HT f x ð t Þg ¼ x ð t Þþ j x ð t Þ ð 3 : 9 Þ

) Z ð f Þ¼ X ð f Þþ j ½ jsgn ð f Þ X ð f Þ¼ X ð f Þ½ 1 þ sgn ð f Þ

) Z ð f Þ¼ 2X ð f Þ;

f 0

ð 3 : 10 Þ

0 ; f \0 :

The analytic signal z(t) associated with the original signal z(t) can be generated
in MATLAB using the command:

z=x+j*hilbert(x)