MWC-System Expander Implementation using Filter Banks Final Presentation

MWC-System

Expander Implementation using Filter Banks

Final Presentation

Supervisors: Professor Yonina Eldar Deborah Cohen

Raz Lifshitz, 052856721Assaf Bismut, 300316684

Goals

- Learn the principles of the Sub Nyquist theory and MWC system

- Comprehend the purpose of the expander

- Comprehend the theory behind the implementation

- Implement the expander using filter banks

- Compare the performance between the old implementation

and the new implementation

MWC Expander Theory ComparisonConclusionsFilter Banks

Goals

Learn the principles of the Sub Nyquist theory and MWC system

Goals MWC Expander Theory ComparisonConclusionsFilter Banks

Goals MWC Expander Theory ComparisonConclusions

MWC – Part I: Analog to Digital ComponentInput: The sparse signal (original signal)Output: M digital channels

Filter Banks

Input: M digital channelsOutput: The final recovered signal


MWC – Part II: DSP Component

Filter Banks


Comprehend the Purpose of the Expander

Filter Banks

The main issue: In order to reconstruct the original signal, M equations are necessary, where every equation is represented by a physical channel. M must fulfill: M>2N, where N is the number of bands of the original signal. Thus, the burden on hardware becomes significant.

Trading channels for sampling rateThe solution: In order to reduce hardware overload, we will combine every “Q” channels into one. This will be done by increasing the bandwidth of each channel by Q. As a result the sampling rate is expedited. For separating back the channels, we will use the Expander.

Preventing burden on hardware

Goals MWC Expander Theory ComparisonConclusionsFilter Banks


ExpanderYi[n]

M channelsYi,Q[n]

Q*M channels

For example, Q=3, in the Frequency domain:

Expander

Yi[n]Xi,1[n]

Xi,2[n]Xi,3[n]

Expander – General Architecture

Filter Banks

Goals MWC ExpanderTheory ComparisonConclusions

Comprehend the theory behind the implementation

Filter Banks

Q

Q

Q

Q


Primary implementation Hybrid implementation

LPF Q LPF Q

Expander Implementations

Filter Banks


Expander ImplementationFilter Banks

Filter Banks


Polyphase - TheoryGiven FIR filter order N

The filter can be written as

Filter Banks


Or

The result is M decimated filters

Filter Banks

Goals MWC ExpanderTheory ComparisonConclusionsFilter Banks

Q

Q

Q

Q


We will observe the case:

Filter Banks - Theory

Filter Banks


We will observe the case:

Filter Banks - Theory

H0 can be written as a sum of its polyphase parts:

Filter Banks

Goals MWC ExpanderTheory Filter Banks ComparisonConclusions

Set it to the Hk (z) phrase (the wanted filter):

DFT of (P0,P1….Pq-1 )

Implement the Expander using filter banks

Goals MWC Expander Theory FilterBanksComparisonConclusions


Expander ImplementationFinal Filter Banks Architecture


Input = IDFT Output

CircularSignalShift

DC

DCFast Frequencies

Fast Frequencies

Fast Frequencies

Output

Low Frequencies

Low Frequencies

Low Frequencies

Low Frequencies

Circular Signal Shift

InitialPhaseFixer


Unlike we assumed in theory, as a result of the analog filter, the first sample of the signal doesn't arrive in t=0.

In the simulator we implement the analog filter by using an digital filter. The order of the digital filter is 5,000.

Initial Phase Fixer


* Looking at one equation

Expander output (Before phase fix) Expected Expander output

InitialPhaseFixer


Initial Phase Fixer

Output Input

InitialPhaseFixer

. . .


Initial Phase problem – solution #2

Another solution to fix the initial phase problem is by setting the analog filter length in a way that : t0 MOD qT =0When t0 is the filter delay


DFT VS FFTProblem: A unit that uses the FFT algorithm to calculate the IDFT can’t provide Wq

kn twiddles because q is an odd number (Therefore q isn’t a power of 2).

Though it is faster to calculate the IDFT of a signal using the FFT algorithm than the direct way. It is negligible because we calculate the IDFT of a short signal (q=3,5…15).

Hardware that calculates the IDFT directly is less common then the ones that uses the FFT algorithm and therefore more expensive, less accurate, and less efficient.


DFT VS FFTA way to use the FFT unit:Though Q is an odd number. The expander can be designed for every number including those in the power of two.

Example: Q=4

Expander (Q=4)(Using FFT)

* Looking at a single channel.

Expander (Q=4)(Using FFT)

L=4 LPF M=3

We can use Interpolation/ Decimation before the expander that uses the FFT unit to get the same output

EXP for q=3



NFAA (No Filter at All)


NFAA (No Filter at All)


Goals MWC Expander Theory FilterBanks ComparisonConclusions

Compare the performance between the old implementation and the new implementation


Quantity of CM (Complexity Multiplies) -Theory 1. Primary: Calculate CM using the fast rate frequency for both LPF’s and the

exponents.2. Hybrid: Calculate CM using the fast rate frequency only for the

exponents. While it use a lower rate to calculate the CM of the LPF’s3. Ours: Calculate CM using the low rate frequency for the LPF, IDFT, and the

exponents.

Table: Number of CM per second Filter Banks Polyphase Primary

M[(L+q2 + q-1)/q] fs(q) ≈ M(L/q)fs(q) M[(q-1)+L]fs(q) ≈ MLfs(q) M[Lq+(q-1)]fs (q)

(*) CM with one was not included, therefore there are only q-1 CM with exponents

M: Hardware Channels ; L: Digital Filter length q: Parameter q of the MWC system ; CM: Complex Multiplies


Quantity of CM (Complexity Multiplies) -Theory

2 4 6 8 10 12 14 160

1

2

3

4

5

6x 10

12 Multiply/Sec

q

Multip

ly/Se

c

HybridOursNFAAPrimary

2 4 6 8 10 12 14 160

0.5

1

1.5

2

2.5

3

3.5

4

4.5x 10

11 Multiply/Sec

q

Multip

ly/Se

c

HybridOursNFAA

Ours Hybrid PrimaryM[(L+q2 + q-1)/q] fs(q) M[(q-1)+L]fs(q) M[Lq+(q-1)]fs (q)

(*) While fs(q) =fs(base) *q ; fs,(base) =2.4e+07

PolyphaseFilter BanksNo Filter

Primary

PolyphaseFilter BanksNo Filter


Quantity of CM (Complexity Multiplies) - Simulation

N = 4


Quantity of CM (Complexity Multiplies) - Simulation


Correlations - Simulation


SNR Simulation

In the case that Mq >2N, there is a 100 % success.For the special case that Mq =2N :

Average of:(q, N, M) = (3,6,4), (5,10,4), (7,14,4)


SNR Simulation

(q, N, M) = (3,6,4) (q, N, M) = (5,10,4)


Primary VS Filter BanksPrimary VS Filter Banks implementation ParameterFilter Banks Imp. requires less Complex Multiplies per second than the Primary imp.

Complex Multiplies per second

Filter Banks imp. has lower execution time than the Primary Imp.

Execution time

Both of the imp. have the same sensitivity to the SNR SNR

Both of the imp. have the same delay Delay

When the signal passes the recovery support stage the correlation of the Filter Banks imp. is slightly better than the primary imp.

Correlation


Appendix

Plot of the polyphase filter:

q = 5

q = 3

q = 7

Documents

MWC-System Expander Implementation using Filter Banks Final Presentation