- Lab #3: - Phase splitter - up conversion and down conversion

1. - 1/40 -
Instructor : Jaewook Kang
At CSNL-GIST
E-mail: jwkkang@gist.ac.kr
2011, Apr. 1st
A Simple Communication System Design
Lab with MATLAB Simulink
- Lab #3: - Phase splitter
- up conversion and down conversion

2. - 2/40 -
Next time…
v Place: IC203
Weeks Time Hour Instructor
1 week
Lab. #0
3.11
(13:00~
16:00)
3
- Overview of Development with Simulink
- QPSK Model with AWGN Channel/ Rayleigh Fading
Channel
- Development Example: Interference Cancellation for Satellite Communi
Junil Ahn
2 weeks
Lab. #1
3.18
(13:00~
14:20)
1.5 - Basic OFDM Junil Ahn
3.18
(14:30~
16:00)
1.5
- Introduction
- How to use Simulink with interleaver
implementation
Jaewook Kang
3 weeks
Lab. #2
3.25
(13:00~
16:00)
3
- How to use S-function builder
- PSF and Matched filter design
- Concept of upsampling and downsampling
Jaewook Kang
4 weeks
Lab. #3
4.1
(13:00~
16:00)
3
- Phase splitter
- Up conversion and down conversion
Jaewook Kang
5 weeks
Lab. #4
4.8
(13:00~
16:00)
3
- How to make subsystem
- Channel implementation using
Jaewook Kang

3. - 3/40 -
Review
v Q1) Why ?
v Q2) What is happen in freq. domain if sampling rate is up ?
v Q3) What about down case?
v Q4)Why PSF ?
v Q5) calculate bandwidth when Rsym=1 sym/sec and L=4.
dw p£

4. - 4/40 -
Today’s main points
v Understand up/down sampling
v Understand characteristic of complex exponential signal
v Learn about Hilbert transform
v Implement Phase splitter using FIR filter
v Understand up/down conversion

5. - 5/40 -
Our target system
vTx part
vRx part
Tx Source Interleaver
QAM
Mapper
PSF X
NCO
↑4
Phase
Splitter
Matched
filter
QAM
DemapperX
NCO
↓4
De-
Interleaver
Rx Source
:Real
:Complex

6. - 6/40 -
up conversion
vWhy upconvert to IF ?
§ 1) To reduce the work of D/A and A/D convertor.
§ 2) To bring the signal only using Inphase channel.

7. - 7/40 -
up conversion
§ As a result, rotational transform of basis in s plane

8. - 8/40 -
up conversion
Re[ ( )] cos(2 ) sin(2 )
Im [ ( )] sin(2 ) cos(2 )
shift shift
shift shift
s t x f t y f t
ag s t x f t y f t
p p
p p
= -
= +

9. - 9/40 -
up conversion
§ When baseband is up-converted to Fs/2
Re[ ( )] cos(2 )shifts t x f tp=
cos(2 ) sin(2 ) cos( )
2 2
s s
s s
f f
x nT y nT x np p p= - =

10. - 10/40 -
up conversion
§ NCO (Numerically Controlled Oscillator ) implementation
§ - Carrier frequency is determined by adjusting ‘k’
§ In general k=4à Then, IF is pi/4
§ - Thus, higher Fs should be required in order to up-convert
baseband into higher band
0
2
s
shift
f
f< £ (s
shift
f
f k
k
= ³is positive number and k 2.0)
2
( ) ( ) ( )
( ) cos(2 ) sin(2 )
1
cos(2 ) cos(2 ) cos(2 )
shift s
s s s
j f nT
s shift s shift s
s
shift s
s
s nT g nT c nT
c nT e f nT j f nT
f n
f nT n
k f k
p
p p
p p p
=
= = +
= =

11. - 11/40 -
up conversion
§ The relationship between up-sampling rate and up-conversion
§ In this case, Fs is sampling rate of baseband signal, g(t)
§ Usually g(t) is the output of PSF, so Fs is up-sampling rate

12. - 12/40 -
Phase splitter
v Purpose of the phase splitter
§ Since Tx only transmit real part of signals, we have to recover
the full complex signal from real part of the received signal.
§ When using FIR filter to implement Phase splitter, we have to
consider the delay of the filter (N-1)/2.
(N- 1)/2 sample
delay
N tap Hilbert Transform
FIR filter
Xr(t) Xr(t)
Xi(t)
Xc(t)

13. - 13/40 -
Phase splitter
v Basic of complex exponential
§ What is shape of ?
§ Can you draw Xc(t) in 3D domain ?
Ø (Hint: real imag, and time axis)
02
c
0
0
fourier transform
x ( )
( ) cos(2 )
( ) sin(2 )
( ) ( ) ( ) ( ) ( ) ( )
j f t
r
i
c r i c r i
Let t Ae
x t A f t
x t A f t
x t x t jx t X f X f jX f
p
p
p
=
=
=
= + ¾¾¾¾¾® = +
02
cx ( ) j f t
t Ae p
=

14. - 14/40 -
Phase splitter
v Basic of complex exponential
§ As t increase, (+) freq. component rotates with CCW and (-) freq.
component rotates with CW.
§ Sine and cosine is the same signal without phase.
§ Real part : even symmetric
§ Imaginary part: odd symmetric
Imag
- f
Re
Acos(2* pi* f0* t)
- f0
f0
Imag
- f
Re
Asin(2* pi* f0* t)
- f0
f0
0
( 0)
cos(2 0)
rx t
A f t Ap
= =
= = 0
( 0)
sin(2 0) 0
ix t
A f tp
= =
= =
Conjugate symmetric

15. - 15/40 -
Phase splitter
v Basic of complex exponential
§ Why cosine and sine are real ?
Ø Imaginary component of (-) and (+) freq. are cancelled each other.
§ Such a fact implies that real and imag components co-exists in
real periodic signal.
Imag
- f
Re
Acos(2* pi* f0* t)
- f0
f0
Imag
- f
Re
Asin(2* pi* f0* t)
- f0
f0
0
( 0)
cos(2 0)
rx t
A f t Ap
= =
= = 0
( 0)
sin(2 0) 0
ix t
A f tp
= =
= =

16. - 16/40 -
Phase splitter
v Basic of complex exponential
§ Phase: The angle between real and imag component.
§ Magnitude: The sum of real part of two freq component.
Ø According to time, the phase and magnitude is changing.

17. - 17/40 -
Phase splitter
v Basic of complex exponential
§ Complex exp. signal has only one frequency component such
that imaginary part exists.
§ How to extract real/imaginary part of Xc(t) ?
02
cx ( ) j f t
t Ae p
=

18. - 18/40 -
Phase splitter
v Hilbert transform
§ Find imaginary part of Xc(t) from real part only when Xc(t) is complex exponential.
Imag
- f
Re
- f0
f0
02j f t
Ae p
Imag
- f
Re
Acos(2* pi* f0* t)
- f0
f0
Re[]
Imag
- f
Re
Asin(2* pi* f0* t)
- f0
f0
Hilbert TR
+
Imag
- f
Re
- f0
f0
02j f t
Ae p
Xc Xr
Xi_hat
Xc_hat
X
j

19. - 19/40 -
Phase splitter
v Transfer function of Hilbert transform
§ The transfer function have sign shape on imaginary axis.

20. - 20/40 -
Phase splitter
v Impulse response of Hilbert transform
§ The discrete form can be obtained by sampling such that
22
2
0
2 22
0
2
( ) ( )
1
[1 cos( )]
s
s
s
s
f
ft
f
f
ft ft
f
s
h t H f e df
je df je df
f t
t
p
p p
p
p
-
-
=
= + -
= -
ò
ò ò
:apply iFT to H(f) of HF
1
[1 cos( )]n
n
p
p
-h(n)=

21. - 21/40 -
Phase splitter
v Implementation of Hilbert transform using FIR filter
§ 1) Determine the number of taps, N
Ø The larger, the nicer. But it cause longer delay.
Ø The delay should be multiple of up sampling rate for synchronization.
Ø Let N
§ 2) Generate h(n) : N=25 à delay =12
§ 3) put h(n) into FIR filter as coefficient.

22. - 22/40 -
Phase splitter
v all clear
v fs=1;% fs must have '1' to get nomalized magnitude
response
v TapNum=73;% the number of FIR filter
v n=0:(TapNum-1);
v delay=(TapNum-1)/2% the delay of FIR filter
v
v % generate Hilbert transform Impulse response
v ht= fs./(pi*(n-delay)).*[1-cos(pi*(n-delay))];
v %alternate form
v %H=2*sin(pi*(n-10)/2).^2./(pi*(n-10))
v
v ht(delay+1)=0;%// by L'Hopital's rule
v
v % Impulse response of hilbert tansform FIR filter
v figure(1);
v stem(ht);
v xlabel('X axis-Time-index (n)','fontsize',12);
v ylabel('Y axis-Magnitude of h(n) ','fontsize',12);
v title('bf{Impulse response of Hilbert tansform,
h(n)}','fontsize',12);
v % obtain the magnitude response of Hilbert transform
v Hf=abs(fft(ht));
v % Magnitude response of hilbert transform FIR
filter
v figure(2);
v stem(Hf);
v xlabel('X axis-Frequency-index (m)','fontsize',12);
v ylabel('Y axis-Magnitude,|H(m)| ','fontsize',12);
v title('bf{Magnitude response of Hibert
transform, |H(m)|}','fontsize',12);
v
v Hf(TapNum-delay+1:TapNum)=Hf(TapNum-
delay+1:TapNum).*-1;
v
v figure(3);
v stem(Hf);
v xlabel('X axis-frequency-index (m)','fontsize',12);
v ylabel('Y axis-imaginary axis, jH(m)
','fontsize',12);
v title('bf{Hilbert transform in frequency
domain}','fontsize',12);

23. - 23/40 -
Phase splitter
v Implementation of Hilbert transform using FIR filter
§ 1) Determine the number of taps, N
Ø The larger, the nicer. But it cause longer delay.
Ø The delay should be multiple of up sampling rate for synchronization.
Ø Let N
§ 2) Generate h(n) : N=25 à delay =12
§ 3) put h(n) into FIR filter as coefficient.

24. - 24/40 -
Next time…
v Place: IC203
Weeks Time Hour Instructor
1 week
Lab. #0
3.11
(13:00~
16:00)
3
- Overview of Development with Simulink
- QPSK Model with AWGN Channel/ Rayleigh Fading
Channel
- Development Example: Interference Cancellation for Satellite Communi
Junil Ahn
2 weeks
Lab. #1
3.18
(13:00~
14:20)
1.5 - Basic OFDM Junil Ahn
3.18
(14:30~
16:00)
1.5
- Introduction
- How to use Simulink with interleaver
implementation
Jaewook Kang
3 weeks
Lab. #2
3.25
(13:00~
16:00)
3
- How to use S-function builder
- PSF and Matched filter design
- Concept of upsampling and downsampling
Jaewook Kang
4 weeks
Lab. #3
4.1
(13:00~
16:00)
3
- Phase splitter
- Up conversion and down conversion
Jaewook Kang
5 weeks
Lab. #4
4.8
(13:00~
16:00)
3
- How to make subsystem
- Channel implementation using
Jaewook Kang