Discrete Fourier Transform and Signal Spectrum - Digital Signal Processing: Fundamentals and Applications - page 98

We can modify the amplitude spectrum to a one-side amplitude spectrum by doubling the amplitudes
in Equation (4.19) , keeping the original DC term at k ¼ 0. Thus we have

8
<

1
N jXð 0 Þj;

k ¼ 0

A k ¼ (4.20)

: 2

N jXðkÞj;

k ¼ 1 ; / ; N= 2

We can also map the frequency bin k to its corresponding frequency as

f ¼ kf s
N

(4.21)

Correspondingly, the phase spectrum is given by

f k ¼ tan 1 Imag ½XðkÞ ; k ¼ 0 ; 1 ; 2 ; / ; N 1 (4.22)

Real ½XðkÞ

Besides the amplitude spectrum, the power spectrum is also used. The DFT power spectrum is
defined as

n
ð Real ½XðkÞÞ

2 o

1
N

1
N

2 2

P k ¼

2 jXðkÞj

¼ þð Imag ½XðkÞÞ ; k ¼ 0 ; 1 ; 2 ; / ; N 1 (4.23)

2

Similarly, for a one-sided power spectrum, we get

8
<

1
N

2

k ¼ 0

2 jXð 0 Þj

P k ¼ (4.24)

: 2

N

2

2 jXðkÞj

k ¼ 0 ; 1 ; / ; N= 2

and

f ¼ kf s
N

(4.25)

Again, notice that the frequency resolution, which denotes the frequency spacing between DFT
coefficients in the frequency domain, is defined as

D f ¼ f s
N

ð Hz Þ (4.26)

It follows that better frequency resolution can be achieved by using a longer data sequence.