Frequency Upconversion and Downconversion
Frequency Upconversion and Downconversion ELEC 433 - Spring 2013 Evan Everett and Michael Wu
Transmit Chain Data
∼ ∼ ∼
D/A Baseband Upconversion
RF Upconversion
∼ ∼ ∼
RF Up/Downconversion •
Purpose: •
Baseband is great for processing
•
RF better for propagation RF Channel
Baseband Tx Processing
Baseband Rx Processing
XC (f )
X(f )
−fC
X(f )
fC
RF Upconversion •
Goal: Convert complex I/Q samples at baseband to real signal centered at carrier
•
Strategy: shift in frequency is multiplication by j2πfc t x(t)e ↔ X(f − fC ) sinusoid in time:
•
Remember the reality condition: If x(t) is real then X(−f ) = X ∗ (f ) X(f )
X ∗ (f + fC )
−fC
XC (f ) X(f − fC )
fC
RF Upconversion Briefly discussed in Modulation lab:
Text
RF Upconversion xI(t) cos(2πƒct) -sin(2πƒct) xQ(t)
xc(t)
RF Upconversion x(t) = xI (t) + jxQ (t)
xC (t) = xI (t) cos(2πfC t) − xQ (t) sin(2πfc t)
xI(t) cos(2πƒct)
xc(t)
-sin(2πƒct)
= |x(t)|cos(2πfC t − ∠x(t)) = Re[x(t)ej2πfC t ]
XC (f ) = X ∗ (f + fC ) + X(f − fC )
xQ(t)
X(f )
X ∗ (f + fC )
−fC
XC (f )
X(f − fC )
fC
RF Downconversion •
Goal: Convert real-valued signal at carrier to I/Q samples at baseband
•
Strategy: shift (multiply by sinusoid) and filter
RF Downconversion 1) Shift by ƒc by multiplying by a complex sinusoid XC (f )
−fC
xc (t)ejπfc t 2fC
fC
2) Lowpass Filter to remove high frequency content X(f )
2fC
RF Up/Downconversion XC (f )
X(f )
xI(t)
−fC
X(f )
fC
cos(2!ƒct)
2fC
LPF
xI(t)
LPF
xQ(t)
xc(t)
cos(2!ƒct) -sin(2!ƒct)
sin(2!ƒct)
xQ(t)
Channel-ready signal Source I/Q signals
Recovered I/Q signals
Transmit Chain •
Data
WARP radios implement RF upconversion and downconversion for us
∼ ∼ ∼
D/A Baseband Upconversion
RF Upconversion
∼ ∼ ∼
Transmit Chain •
Data
•
WARP radios implement RF upconversion and downconversion for us
∼ ∼ ∼
D/A
WARP Radio
Baseband Upconversion
This week you will be implementing digital baseband up/downcoversion
Baseband Up/Downconversion •
Reason: most radios reject DC input due to an RF issue called DC offset (a.k.a. carrier leakage) AC coupled I/Q inputs
I Radio Q
•
Goal: shift I/Q samples away from DC
•
Difference from RF upconversion: output is still complex (I/Q)
Baseband+RF Upconversion •
Radio’s usable input spectrum has a notch at DC
•
Shift I/Q samples to an intermediate frequency
!
fIF
fIF
DC
via FPGA
IF
fRF
via Radio
RF
Baseband Up/Downconversion
fIF
xI(t)
upconverted xI(t)
cos(2!ƒct)
cos(2!ƒct)
sin(2!ƒct)
-sin(2!ƒct)
xQ(t)
xI(t)
xQ(t)
upconverted xQ(t)
Complex Multiplication
Complex Multiplication
A Few Complications upconverted xI(t)
xI(t) cos(2!ƒct)
cos(2!ƒct)
sin(2!ƒct)
-sin(2!ƒct)
xQ(t)
•
upconverted xQ(t)
xI(t)
xQ(t)
Desired symbol rate = 5 Mhz, DAC sample clock = 40Mhz •
Before upconversion, upsample I/Q signals by 8 then filter (we know how to build efficient interpolaters).
•
Do opposite on the receiver side.
•
Delay between Tx and Rx: what if the receiver starts listening after a transmitter begins sending?
•
Tx/Rx carrier frequency mismatch: residual frequency left at receiver
Transmit Chain Data
∼ ∼ ∼
D/A Baseband Upconversion
RF Upconversion
∼ ∼ ∼
RF Up/Downconversion •
Purpose: •
Baseband is great for processing
•
RF better for propagation RF Channel
Baseband Tx Processing
Baseband Rx Processing
XC (f )
X(f )
−fC
X(f )
fC
RF Upconversion •
Goal: Convert complex I/Q samples at baseband to real signal centered at carrier
•
Strategy: shift in frequency is multiplication by j2πfc t x(t)e ↔ X(f − fC ) sinusoid in time:
•
Remember the reality condition: If x(t) is real then X(−f ) = X ∗ (f ) X(f )
X ∗ (f + fC )
−fC
XC (f ) X(f − fC )
fC
RF Upconversion Briefly discussed in Modulation lab:
Text
RF Upconversion xI(t) cos(2πƒct) -sin(2πƒct) xQ(t)
xc(t)
RF Upconversion x(t) = xI (t) + jxQ (t)
xC (t) = xI (t) cos(2πfC t) − xQ (t) sin(2πfc t)
xI(t) cos(2πƒct)
xc(t)
-sin(2πƒct)
= |x(t)|cos(2πfC t − ∠x(t)) = Re[x(t)ej2πfC t ]
XC (f ) = X ∗ (f + fC ) + X(f − fC )
xQ(t)
X(f )
X ∗ (f + fC )
−fC
XC (f )
X(f − fC )
fC
RF Downconversion •
Goal: Convert real-valued signal at carrier to I/Q samples at baseband
•
Strategy: shift (multiply by sinusoid) and filter
RF Downconversion 1) Shift by ƒc by multiplying by a complex sinusoid XC (f )
−fC
xc (t)ejπfc t 2fC
fC
2) Lowpass Filter to remove high frequency content X(f )
2fC
RF Up/Downconversion XC (f )
X(f )
xI(t)
−fC
X(f )
fC
cos(2!ƒct)
2fC
LPF
xI(t)
LPF
xQ(t)
xc(t)
cos(2!ƒct) -sin(2!ƒct)
sin(2!ƒct)
xQ(t)
Channel-ready signal Source I/Q signals
Recovered I/Q signals
Transmit Chain •
Data
WARP radios implement RF upconversion and downconversion for us
∼ ∼ ∼
D/A Baseband Upconversion
RF Upconversion
∼ ∼ ∼
Transmit Chain •
Data
•
WARP radios implement RF upconversion and downconversion for us
∼ ∼ ∼
D/A
WARP Radio
Baseband Upconversion
This week you will be implementing digital baseband up/downcoversion
Baseband Up/Downconversion •
Reason: most radios reject DC input due to an RF issue called DC offset (a.k.a. carrier leakage) AC coupled I/Q inputs
I Radio Q
•
Goal: shift I/Q samples away from DC
•
Difference from RF upconversion: output is still complex (I/Q)
Baseband+RF Upconversion •
Radio’s usable input spectrum has a notch at DC
•
Shift I/Q samples to an intermediate frequency
!
fIF
fIF
DC
via FPGA
IF
fRF
via Radio
RF
Baseband Up/Downconversion
fIF
xI(t)
upconverted xI(t)
cos(2!ƒct)
cos(2!ƒct)
sin(2!ƒct)
-sin(2!ƒct)
xQ(t)
xI(t)
xQ(t)
upconverted xQ(t)
Complex Multiplication
Complex Multiplication
A Few Complications upconverted xI(t)
xI(t) cos(2!ƒct)
cos(2!ƒct)
sin(2!ƒct)
-sin(2!ƒct)
xQ(t)
•
upconverted xQ(t)
xI(t)
xQ(t)
Desired symbol rate = 5 Mhz, DAC sample clock = 40Mhz •
Before upconversion, upsample I/Q signals by 8 then filter (we know how to build efficient interpolaters).
•
Do opposite on the receiver side.
•
Delay between Tx and Rx: what if the receiver starts listening after a transmitter begins sending?
•
Tx/Rx carrier frequency mismatch: residual frequency left at receiver
