The following algorithms have been tested as a method of converting an I.F. from 16 to 512 MHz to 31 real 16 MHz USB channels:

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1 DBE Memo#1 MARK 5 MEMO #18 MASSACHUSETTS INSTITUTE OF TECHNOLOGY HAYSTACK OBSERVATORY WESTFORD, MASSACHUSETTS 1886 November 19, 24 Telephoe: Fax: To: From: Mark 5 Developmet Group Ala E.E. Rogers ad Has F. Hiteregger Subject: Real to real polyphase filter bak A] Method usig overlapped FFTs The followig algorithms have bee tested as a method of covertig a I.F. from 16 to 512 MHz to 31 real 16 MHz USB chaels: 1] Sample the iput data with a 124 MHz clock covertig the 8 to 54 or 52 to 116 MHz I.F. bad from aalog to digital. 2] Put the samples xt [ ] i blocks of 64 samples ito a series of 64 m tap FIR subfilters so that the output of the th subfilter is where [ ] = [ ] [ + 64 ] y b x k b h k = sub-filter umber b = data block umber ad ito aother set of subfilters with a delay of 32 so that [ ] [ ] z ( b) = x + 64k b h + 64k 3] Move the y outputs ito the real iputs of a 64 poit complex FFT ad the z outputs ito the complex iputs Re X = y b [ ] [ ] [ ] = [ ] Im X z b This allows the FFT to trasform 2 real sigals with oe complex FFT. 4] For block b= take 64 poit FFT 1

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3 5] Combie the outputs Y[ ] of the FFT to separate the trasforms of the first 64 samples put i the real iputs from those put ito the imagiary iputs [ ] = [ ] + [ ] [ ] = [ ] [ ] Re Z Re Y Re Y Re Z = Re Y + Re Y 64 for > Im Z Im Y Im Y Im Z = Im Y Im Y 64 for > ad Re Z = Im Y + Im Y Re Z = Im Y + Im Y 64 for > Im Z = Re Y Re Y Im Z = Re Y 64 Re Y for > 6] Form the real output sequeces for the 31 output chaels as follows: [ ] b[ ] [ 2 + 1] = Im [ ] C 2b = s Re Z for < < 32 C b ss Z 1 where s = 1 for eve blocks s = -1 for odd blocks s 1 = 1 for eve chaels s 1 = -1 for odd chaels The = chael is ot useful due to aliasig. b 7] Form the ext set of samples for each chael from the ext block b=1 etc. The FIR filter coefficiets ca be obtaied from samplig a widowed sic fuctio as follows: ( ) ( ) [ ] = π ( ) where w= ( i N 2) π 64 hi.54.46cos 2 i N 1 sic w N = m 64 B] Method usig o overlapped FFTs 1] Sample iput with 124 MHz clock 2] Put samples ito FIR subfilter 3

4 ad [ ] = [ ] [ + 64 ] y b x k b h k m= m= [ ] [ ] z ( b) = x + 64k b h + 64k 3] Put ito 64 poit FFT 4] Take FFT [ ] = [ ] [ ] = [ ] Re X y b Im X z b 5] Combie outputs to separate results [ ] = [ ] + [ ] [ ] = [ ] [ ] [ ] = [ ] + [ ] [ ] = [ ] [ ] Re Z Re Y Re Y 64 Im Z Im Y Im Y 64 Re Z Im Y Im Y 64 Im Z Re Y 64 Re Y 6] Form output sequece of 4 samples per 128 sample block for each chael [ 4 ] = Re b[ ] [ 4 + 1] = [ 2 ] [ 4 + 2] = Re b [ ] [ 4 + 3] = [ 2 + 1] C b Z C b I b C b Z C b I b where I [ ] are iterpolated values from Im Z ad Im Z formed by takig the sequece Q[k] ad applyig a FIR iterpolatio filter [ ] = [ ] Qb Im Zb Q 2b+ 1 = Im Z [ ] = [ + 2] [ ] Ib Qb k m Hk b (.7.3cos( 2 1 )) sic( ( 2.5) π) The iterpolatio FIR has values of [ ] π ( ) H i = i N i N + 16 taps are sufficiet to provide more tha 4 db image rejectio across 8% of each filter output. 4

5 C] Method usig mixig before the FFT A alterate method is to perform SSB mixig operatios prior to eterig the FFT. Followig the method of Crochiere ad Rabier s figure 7.48: 1] Sample the iput with A/D 2] Put the samples x[t] ito a series of 32 m tap complex subfilters m 2 1 [ ] = [ ] [ + 64 ] y b x k b h k s m 2 1 [ ] = [ ] [ ] z b x k b h k s where s = 1, -1, 1, -1 etc. i this case ( ) ( ) [ ] = π ( ) where w= ( i N 2) π 64 hi.54.46cos 2 i N 1 sic w N = 32m 3] Perform a complex rotatio of each output of FIR sub-filters before puttig ito the FFT. [ ] = ( [ ] + [ ]) X y b iz b e b 2π i 128 4] The real outputs of the FFT form the real samples for each chael. The output chael order i order of icreasig frequecy without resortig will be, 31,1, 3,2, 29, 3, 28, 4, 27, 5, 26, 6, 25, 7, 24, 8, 23, 9, 22, 1, 21, 11, 2, 12, 19, 13, 18, 14, 17, 15, 16 ad after resortig the chaels alterate betwee USB ad LSB. Figure 1 shows the 32 output filter resposes from the secod method usig widowed sic symmetric FIR. [More work will be eeded to optimize the filter desig.] The badpasss fuctio for the methods A&B are similar ad are show i Figure 2. The first chael is ot useful but sice this chael is at the badpass ed it is likely that it would be uusable ayway owig to the shape of the aalog badpass filter ahead of the A/D. 5

6 Figure 1. Chael badpasses from method C Figure 2. Chael badpass from methods A ad B Compariso of computatio required for 3 methods: A.) Real to real filter bak usig overlapped FFTs This is the first method described i this memo. It uses overlapped FFTs to provide iterpolated outputs so that the complex outputs ca be shifted to positive frequecys by multiplicatio by a vector that rotates by 9 degrees per output sample ad subsequet droppig of the imagiary part. For efficiecy oe 64 poit FFT serves double duty by processig both the primary ad the overlapped FFTs. 6

7 B.) Real to real filter bak usig iterpolatio of outputs This method is similar to method A except that the FFTs are ot overlapped so that 1 double duty 64 poit FFT services 128 iput samples. Every other output sample eeds to be time shifted by iterpolatio C.) Real to real usig SSB mixig prior to FFT This is the third method described i this memo. For the details of this method we refer to Crochiere ad Rabier. This method has the advatage of providig all 32 filtered outputs. The followig table compares the umber of multiplicatios eeded for each algorithm. This real to real method requires about twice the umber of multiplicatios eeded by the complex to complex method. D.) Complex to complex filter bak The simplest polyphase filter bak is oe which coverts complex samples iput to complex outputs. A 32 poit FFT is eeded to obtai 32 filtered outputs. We take this as our referece method. However for VLBI we require real outputs. I/O Overlap FFT Sample rate Gs/s FFT size Multiplies per 64 s FFT FIR PFB FIR INTR Iput Output filters Subfilter Tot Size m Size m A Real/real Yes Noe B Real/real Noe C Real/real Noe Noe D Cmpx/cmpx Noe Noe Table. Polyphase filter bak for 32 output chaels for 5 MHz badwidth Notes: 1) Assumes 2log2 multiplies for a poit FFT 2) The FIR, assumed for the complex filter bak these calculatios has 16 taps per sub-filter or a total of 512 taps. The FIRs for the real to real cases are chose to have the same shape factors as the complex to complex case. 3) Method B is similar to method A but there is o overlap so that 1 64 poit FFT services 128 iput samples. The extra output samples required to allow coversio to real are obtaied usig a FIR filter o each output to iterpolate betwee output samples. Oly the imagiary outputs eed to be iterpolated ad the filter is symmetric so that oly half the umber of multipliers are eeded. 7