Re-indexing the DFT (n and k) We can investigate the various - - PowerPoint PPT Presentation

Re-indexing the DFT (n and k)

• We can investigate the various

implementations of the DFT by looking at ways to re-index values of n and k.

• Idea:
• Using properties of addition and

multiplication we can rewrite the equation. 𝑌 𝑙 = σ𝑜=0

𝑂−1 𝑦 𝑜 𝑓−𝑘2𝜌𝑙𝑜/𝑂 = σ𝑜=0 𝑂−1 𝑦 𝑜 𝑋 𝑂 𝑙𝑜

• Notice that the sum of n from 0 to N-1 can

take on any sequence as long as all the integer terms exist are included only once.

• Furthermore, these terms can include

multiples of N as WN

kn+mN = WN kn.

• For these cases, WN

kn is replace with:

WN

kn mod N

Bluestein Chirp-z Transform

• Rewrite the term kn as

nk = n2/2 + k2/2 – (k-n)2/2

• Check

= n2/2 + k2/2 – n2/2 - k2/2 + nk = nk

• The term we are interested in is k-n.
• Rewrite the term kn as

𝑌 𝑙 = σ𝑜=0

𝑂−1 𝑦 𝑜 𝑓−𝑘2𝜌𝑙𝑜/𝑂

𝑌 𝑙 = σ𝑜=0

𝑂−1 𝑦 𝑜 𝑋 2𝑂 𝑜2+𝑙2− 𝑙−𝑜 2

𝑌 𝑙 = 𝑋

2𝑂 𝑙2 σ𝑜=0 𝑂−1 𝑦 𝑜 𝑋 2𝑂 𝑜2 𝑋 2𝑂 − 𝑙−𝑜 2

Bluestein Chirp-z Transform

• The interpretation of this equation.

𝑌 𝑙 = 𝑋

2𝑂 𝑙2 ෍ 𝑜=0 𝑂−1

𝑦 𝑜 𝑋

2𝑂 𝑜2 𝑋 2𝑂 − 𝑙−𝑜 2

Pre-multiply by a chirp Convolution (FIR filter) k:[0:N-1], n:[0:N-1] FIR filter length [-(N-1):N-1]=2N-1 Post-Multiply by a chirp

Bluestein Chirp-z Transform

𝑌 𝑙 = 𝑋

2𝑂 𝑙2 ෍ 𝑜=0 𝑂−1

𝑦 𝑜 𝑋

2𝑂 𝑜2 𝑋 2𝑂 − 𝑙−𝑜 2

CORDIC z-N 2N-1 Tap FIR

2 complex multipliers 8 Multipliers 4 adders 2N-1 tap complex filter. (2N-1)*4 Multipliers (2N-1)*2 Adders Take advantage of a symmetrical filter N*4 Multipliers (N-1) + N Adders ~= 2N Adders Latency 2*N

Counter phase Counter freq

N-Point Complex DFT (even)

z-1

WN

0: (from ROM)

X[0] In Parallel Adders: N*4. Multipliers: N*4. Registers: N. Latency: N.

z-1

WN

1

X[1]

z-1

WN

N-1

X[N-1] ...

n-counter

rst rst rst

Data In Memory Data Out Memory

??? ena ???

Chirp-z Length 8

• Matlab Implementation

Chirp-z Length 16 sim

Chirp-z Length 16 sim

Chirp-z Length 16 sim

Chirp-z Length 128 sim

Chirp-z Length 128 sim

Prime Number Index Generator

• For a prime number (p), we can use a

generator (another prime g) to create a integer sequence that will contain each value in 1 to (p-1).

• a = gα mod p (for α=0:N-2)
• Example: p=7, g=3
• 30 mod 7 = 1
• 31 mod 7 = 3
• 32 mod 7 = 2
• 33 mod 7 = 6
• 34 mod 7 = 4
• 35 mod 7 = 5
• Let’s us complete a sum over an index

with a substitution of variables that includes a exponent.

Rader Transform

𝑌 𝑙 = ෍

𝑜=0 𝑂−1

𝑦 𝑜 𝑓−𝑘2𝜌𝑙𝑜/𝑂

• Re-index n using the index generator

𝑌 𝑙 = ෍

𝑜=0 𝑂−2

𝑦 𝑕𝑜 mod 𝑂 𝑋

𝑂 𝑙 𝑕𝑜 mod 𝑂 + 𝑦 0

• Mod N is essentially multiplying by 1.

𝑌 𝑙 = ෍

𝑜=0 𝑂−2

𝑦 𝑕𝑜 mod 𝑂 𝑋

𝑂 𝑙𝑕𝑜 + 𝑦 0

Rader Transform

𝑌 𝑙 = ෍

𝑜=0 𝑂−1

𝑦 𝑜 𝑓−𝑘2𝜌𝑙𝑜/𝑂

• Re-index k using the index generator

𝑌 𝑙 = ෍

𝑜=0 𝑂−2

𝑦 𝑕𝑜 mod 𝑂 𝑋

𝑂 𝑙𝑕𝑜 + 𝑦 0

𝑌 𝑕𝑙 mod 𝑂 = ෍

𝑜=0 𝑂−2

𝑦 𝑕𝑜 mod 𝑂 𝑋

𝑂 𝑕𝑙 mod 𝑂 𝑕𝑜

+ 𝑦 0 *Note: X[0] needs to be calculated directly. 𝑌 𝑕𝑙 mod 𝑂 = ෍

𝑜=0 𝑂−2

𝑦 𝑕𝑜 mod 𝑂 𝑋

𝑂 𝑕𝑙𝑕𝑜 𝑛𝑝𝑒 𝑂 + 𝑦 0

𝑌 𝑕𝑙 mod 𝑂 = ෍

𝑜=0 𝑂−2

𝑦 𝑕𝑜 mod 𝑂 𝑋

𝑂 𝑕𝑙+𝑜 𝑛𝑝𝑒 𝑂 + 𝑦 0

Rader Transform: Interpretation

𝑌 𝑕𝑙 mod 𝑂 = ෍

𝑜=0 𝑂−2

𝑦 𝑕𝑜 mod 𝑂 𝑋

𝑂 𝑕𝑙+𝑜 + 𝑦 0

• The k+n term can be seen as a correlation, which

can be accomplished with an FIR filter.

• Process:
• 1. Reorder x[n] using index generator.
• 2. Preform correlation with an FIR filter.
• 3. Directly add x[0].
• 4. Put the order back.
• 5. X[0] is determined independently as sum(x[n]).

Matlab Simulation

simulation

simulation

More points: N=59, g=2

More points: N=59, g=2

Resources Utilization

• Goertzel, Chirp-Z, and Rader.
• All use similar resources and

latencies.

• Can be used in a serial or parallel

architecture.

• Perform the convolution in a point-by-

point fashion.

• Similar trades in resource vs latency.

DSN03

• Hints
• Implement the routine in Matlab to

generate intermediate values.

• May not be exactly the same due to rounding

errors.

• Make matlab simulation as similar as possible.
• Start small.
• Get a single point working for a value of N=4.
• Get all points working for small N.
• Make use of simulation waveforms.
• Set up the waveforms in a way that you can

easily compare with expected results (matlab).

• Save your simulation configuration so you don’t

have to set up radix and analog ... Every time.