Digital Signal Processing Reference
In-Depth Information
INPUT DATA for SPECTRAL PARAMETERS of a SYNTHESIZED TIME SIGNAL or FID
Argand Plot for Input Complex Frequencies
ν
k
: x
Absolute Values of Input Amplitudes |d
k
| : x
0
0.25
13
21
19
18
6
11
B
0
=1.5T
d
k
= |d
k
|e
i
φ
k
≡
|d
k
| ,
φ
k
= 0 (k = 1,2,...,25)
15
10
22
16
7
24
20
0.05
0.2
12
9
8
5
14
23
17
6
2
0.1
0.15
4
3
25
1
H
2
O
7
15
Lip
0.15
16
0.1
8
3
9
1
H
2
O
22
17
14
12
Lip
25
5
0.2
0.05
20
19
10
4
21
24
B
0
=1.5T
23
18
11
0.25
2
13
0
5
4
3
2
1
5
4
3
2
1
(i) Re(
ν
k
) (ppm)
(iv) Chemical shift (ppm)
Argand Plot for Input Signal Poles z
−1
k
Argand Plot for Input Signal Poles z
k
0
0
25
25
B
0
=1.5T
Arc of the unit circle ( line)
B
0
=1.5T
24
24
−0.2
0.2
23
22
Arc of the unit circle ( line)
Input harmonic variables (dots)
23
21
22
20
21
−0.4
0.4
19
20
18
z
k
= e
2i
πτ
ν
k
19
17
|z| < 1
16
18
14
15
17
−0.6
0.6
|z| < 1
16
13
12
15
10
11
Input
harmonic
variables (dots)
z
−1
14
9
13
8
7
−0.8
12
0.8
6
5
4
3
11
2
10
1
9
|z| > 1
8
7
6
−1
5
1
k
= e
−2i
πτ
ν
k
, |z| > 1
1
4
2
3
1
0.75
0.5
0.25
0
1
0.75
0.5
0.25
0
(v) Re(z
−1
k
(ii) Re(z
k
)
)
K
d
k
z
n
k
, z
k
= e
2i
πτ
ν
k
(K = 25)
K
d
k
z
n
k
, z
k
= e
2i
πτ
ν
k
(K = 25)
Time Signal: c
n
=
Σ
k=1
Time Signal: c
n
=
Σ
k=1
2
2
Real part of time signal c
n
or FID
Imaginary part of time signal c
n
or FID
1.5
1.5
B
0
=1.5T
B
0
=1.5T
1
1
0.5
Bandwidth = 1000 Hz
0.5
Bandwidth = 1000 Hz
0
0
−0.5
N = 1024,
τ
= 1 ms
−0.5
N = 1024,
τ
= 1 ms
−1
−1
0
250
500
750
1000
1250
0
250
500
750
1000
1250
(iii) Time n (in units of sampling rate
τ
)
(vi) Time n (in units of sampling rate
τ
)
FIGURE 3.1
Argand plot for the input complex frequencies ν
k
and a graphical display of
the absolute values of amplitudes d
k
=|d
k
|: panels (i) and (iv). Argand plots
for the input signal poles or harmonic variables z
k
and their inverses z
−
k
:
panels (ii) and (v). Real Re(c
n
) and imaginary Im(c
n
) part of the input time
signal{c
n
}or FID as a sum of 25 damped complex exponentials with constant
amplitudes d
k
: panels (iii) and (vi).
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