8-1b we see a collection of sine wave harmonics as deÞned in Fig.. These sine wave harmonics are the Fourier series constituents of the symmetrical square wave in Fig.. These − cosine wa
Trang 10 16 32 48 64 80 96 112 128
0
−1
1
x(n)
n
(a)
N := 128 n := 0, 1 N
k := 0, 1 N
x(n) := 0
−1 if n >
1 if n> 0
0 if n = N
0 if n=
N
2
N
2
(c)
XH(k) := −j⋅X(k) if k < N
2
N
2
0 if k=
N 2
j⋅X(k) if k >
−1
−0.5
0
0.5
1
Im(X(k))
k
(b)
Imaginary
X(k) := 1 ∑nN−1= 0 N
n N x(n) ⋅exp −j⋅2⋅π⋅ ⋅k
⋅
Figure 8-1 Example of the Hilbert transform.
Trang 20 16 32 48 64 80 96 112 128
−1
−0.5
0
0.5
1
Re(XH(k))
k
Real
(d )
−4
−3
−2
−10
1
2
3
4
(xh(n))
x(n)
n
(f )
xh1(n) : = 0.25⋅xh(n − 1) + 0.5⋅xh(n) + 0.25⋅xh(n + 1) xh2(n) : = 0.25⋅xh1(n − 1) + 0.5⋅xh1(n) + 0.25⋅xh1(n + 1)
(g)
−4
−3
−2
−10
1
2
3
4
xh2(n)
x(n)
n
(h)
(e)
N (XH(k)) ⋅exp j⋅2⋅π⋅ ⋅n
Figure 8-1 (continued)
Trang 3chapter we will continue to use DFT and IDFT and stay focused on the main objective, understanding the Hilbert transform
Why do the samples in Fig 8-1f and h bunch up at the two ends and
in the center to produce the large peaks? The answer can be seen by comparing Fig 8-1b and d In Fig 8-1b we see a collection of (sine) wave harmonics as deÞned in Fig 2-2c These sine wave harmonics are the Fourier series constituents of the symmetrical square wave in Fig 8-1a In Fig 8-1d we see a collection of (− cosine) waves as deÞned
in Fig 2-2b These (− cosine) wave harmonic amplitudes accumulate at
the endpoints and the center exactly as Fig 8-1f and h verify As the harmonics are attenuated, the peaks are softened The smoothing also tends to equalize adjacent amplitudes slightly The peaks in Fig 8-1h rise about 8 dB above the square-wave amplitude, which is almost always too much There are various ways to deal with this One factor is that the square-wave input is unusually abrupt at the ends and center Smoothing
(equivalent to lowpass Þltering) of the input signal x(n), is a very useful
approach as described in Chapter 4 This method is usually preferred in circuit design
It is useful to keep in mind, especially when working with the HT,
that the quadrature of θ, which is θ ± 90◦, is not always the same thing
as the conjugate of θ If the angle is +30◦, its conjugate is−30◦, but its quadrature is +30◦± 90◦= +120◦ or −60◦ The HT uses θ◦± 90◦ For example, Fig 8-1b shows+j0.5 at k = 127 The HT multiplies this by + j
to get a real value of− 0.5 in part (d) at k = 127 This is a quadrature positive phase shift at negative frequency k= 127 A similar event occurs
at k = 1 Also, at k = 0 and N the phase jump is 180◦ from± j to ∓j, and the same, although barely noticeable, at N /2 (Use a highly magniÞed
vertical scale in Mathcad.)
A different example is shown in Fig 8-2 The baseband signal in part (b) is triangular in shape, and this makes a difference The abrupt changes in the square wave are gone, the baseband spectrum (d) contains only real cosine-wave harmonics, and the Hilbert transform (f) contains
only sine-wave harmonics The sharp peaks in xh(n) that we see in
Fig 8-1g disappear in Fig 8-1h However, for equal peak-to-peak ampli-tude, the square wave of Fig 8-1 has 4.8 dB more average power than the triangular wave of Fig 8-2 It is not clear what practical advantage
Trang 40 16 32 48 64
n
−4
−2
0 2
x(n)
−0.5
0.5
0
1 1.5
Re(X(k))
x(n):=
N:= 128 n := 0, 1 N k := 0, 1 N
3 if n= 0
3 if n = N
(a)
(b)
k
(d )
(c )
∑N−1
n = 0
X(k) := 1
N
−j⋅2⋅π⋅ ⋅k
12·n N
−9 + 12·n if n >
N N
2
⋅
Figure 8-2 Hilbert transform using a triangular waveform.
the triangular wave would have The importance of peak amplitude limits and peak power limits in circuit design must always be kept in mind
BASIC PRINCIPLES OF THE HILBERT TRANSFORM
There are many types of transforms that are useful in electronics work The DFT and IDFT are well known in this book because they transform back
Trang 5xH(k) := −j ⋅ X(k) if k <
0 if k=
1
0.5
0
(e)
N
2
j ⋅ X(k) if k > N
2
N
2
−1
−0.5
4
−4
−3
−2
−10
1
2
3
0 16 32 48 64
k
(f )
(g)
80 96 112 128
0 20 40
n (h)
60 80 100 120
Im(XH(k))
x(n)
Re(xh(n))
∑N−1
n = 0
xh(n) := (XH(k)⋅exp k
N
j ⋅2⋅π⋅ ⋅k
Figure 8-2 (continued)
and forth between the discrete time-domain signal x(n) and the discrete frequency-domain spectrum X (k).
Another very popular transform converts a linear differential equation into a linear algebraic equation For example, consider the differential