1. Hilbert Transforms
A couple of quick notes for
understanding Field II signal processing

2. Why use Hilbert
Transforms?
 Very useful with bandpass applications
 For example, ultrasound signal processing
 Benefits:
 Mathematical basis for representing
bandpass signals
 Easy determination for signal envelope
 May reduce ADC sampling rates

3. Mathematical Basis
1
 Hilbert Transform for signal x(t) x( t ) = x( t ) ∗
ˆ
πt
 NOTE: The HT is a function of time
x (τ )
∞
1
The Definition is actually a
= ∫∞t − τ dτ

convolution! π −
 We get much better insight into the HT
from working in the frequency domain.
+90
 Frequency domain HT transfer
function: − j , when f > 0 f
 
 
H ( f ) = − j sgn ( f ) = + j , when f < 0 -90
 0, when f = 0 
  Phase response

4. Bandpass Signals
 Bandpass signals
 Let z(t) be a bandpass signal
centered around some f0
 z (t) can be expressed as
z ( t ) = x( t ) cos( 2π f ot ) − y ( t ) sin ( 2π f ot )
 In this expression, x(t) and y(t) are
lowpass. z(t) can be written as
z ( t ) = a( t ) cos( 2π f ot + θ ( t ) )
where a( t ) = + x ( t ) + y ( t )
2 2
and θ ( t ) = sin −1 ( y ( t ) / x( t ) )

5. Mathematical Basis
Definition of analytic signal (or
() () ()

pre-envelope) x+ t = x t + j x t ˆ
 Upper graph shows an FFT of the
original signal, x(t).
 Take an FFT of its analytical
counterpart - lower graph
 This can be readily understood by
looking at the analytic signal in the
frequency domain:
2Z ( f ) , when f > 0
 
Z+ ( f ) = Z ( f ) + j[ − j sgn ( f ) ] Z ( f ) =  Z ( 0 ) , when f = 0 
 0, when f < 0 
 

6. Getting the Envelope
 We can now find
the envelope of a
bandpass signal.
 This envelope is
used to form a B-
mode image.
 Most scanners
today use this type
of envelope
detection

7. Getting the Envelope
 We can now find
the envelope of a
bandpass signal.
 This envelope is
used to form a B-
mode image.
 Most scanners
today use this type
of envelope
detection