# 0.2 Derivation of the equations for a basic fdm-tdm transmux  (Page 10/10)

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## Relationship of the filter bank approach to digital spectrum analysis

A matter of confusion to many engineers is that the filter bank scheme seems to produce time samples from FFT bins. This confusion has its root in the fact that the DFT, and hence the FFT, are usually discussed in the context of digital spectrum analysis and are typically spoken of as methods of converting from the time domain to the frequency domain. How then can a DFT-based filter bank produce time samples from spectral bins? In fact, the right perspective is the opposite one.

Consider again [link] from the viewpoint of digital spectrum analysis. A simple FFT-based spectrum analyzer accepts N samples of the input sequence, weights or windows the data, transforms it by using an N-point FFT, and then estimates the power spectral density by computing the magnitude square of the bin outputs. Comparing these steps to [link] , we see that they are identical except for two things: (1) the magnitude squaring operation at the bin outputs and (2) the fact that in spectrum analysis the window and transform operation is rarely done for every input sample. (Typically it is done every N-th sample [called 1:1 overlapping ] or even less frequently.) These facts suggest that DFT/FFT-based digital spectrum analysis is derived from the filter bank concept rather than the other way around. The filter bank shown in [link] uses a transform computed over a record of weighted, delayed input data to split the input signal's energy into N spectral bands. The degree to which this separation is completed depends on the choice of windowing or weighting function and on the length of the transform. If the function is chosen properly, the windowing operation and the DFT/FFT computation can be computed less frequently, that is, decimation can be introduced. In this context, the simple FFT-based spectrum analyzer can be recognized to perform an instantaneous power measurement at the output of each of the filters in the bank. The quality of the analyzer depends on the window function chosen and the DFT/FFT order N (as they affect the passband shape), the rate at which the filter outputs are computed (given by the decimation factor M ), and the number of instantaneous power measurements averaged to obtain the spectral estimate. Thus we can conclude that the digital spectrum analyzer approximates the true power spectrum by measuring the power seen in each of the bandpass filter outputs produced by the DFT-based filter bank.

An interesting sidelight is that the most common name for the transmultiplexer preprocessor stems from the filter bank's relationship with digital spectrum analysis. Look again at [link] . The input to the QN-point FFT is $QN$ weighted and delayed input samples. From the bank-of-tuners viewpoint we know that the weighting function $w\left(k\right)$ is just the tuner pulse response $h\left(k\right)$ needed to bandlimit the tuned signal properly. In the context of spectrum analysis, however, this function is called a data window. They are in fact identical and the tuner viewpoint provides the analytical basis on which to design the needed window. We've already observed that after pruning the FFT, the QN-point transform separates into two sections. The first section folds together Q windowed samples at a time to generate the N-point input to the FFT. From this viewpoint, it is commonly referred to as the window-and-fold section of the FDM-to-TDM transmultiplexer.

## Stylized fortran implementation of a basic fdm-tdm transmux

Table 1 shows a stylized example of a software implementation of an FDM-TDM transmultiplexer. Some details of the initialization steps have been blurred for the sake of simplicity and the parameters user are certainly not those appropriate to all applications, but the code should serve as an accurate guide to the amount of computation needed and its organization.

SUBROUTINE TMUX(INPUT_ARRAY, INPUT_POINTER, OUTPUT_VECTOR) CC SUBROUTINE TMUX - IMPLEMENTS A BASIC FDM-TO-TDM TRANSMUX. C A NEW VECTOR OF CHANNELIZED CHANNEL OUTPUTS, CALLEDC "OUTPUT_VECTOR" IS PRODUCED EACH TIME THE SUBROUTINE IS CALLED, C UNLESS THE DATA IN "INPUT_ARRAY" IS EXPENDED.C PARAMETER N = 64 !NUMBER OF CHANNELS AND/OR BINSPARAMETER Q = 3 !WEIGHTING FUNCTION EXPANSION FACTOR PARAMETER M = N !DECIMATION FACTORC THIS CHOICE OF M YIELDS BASIC TRANSMUX CINTEGER M, N, Q, INPUT_POINTER, J, K, INDEX COMPLEX INPUT_ARRAY(1), OUTPUT_VECTORS(N), VRP(N)REAL WEIGHTING(N*Q) CDATA WEIGHTING/ * QN values of the weighting function h(k) */ CC ****************** MOVE TIME POINTER ************************** CINPUT_POINTER = INPUT_POINTER + M CC ****************** COMPUTE v(r,p) ***************************** CDO 10 J=1,N 10 VRP(J) = CMPLX(0.,0.) !ZERO THE VECTOR VRPC DO 30 J=1,NDO 20 K=1 Q INDEX = (K-1)*N+J-1 !COMPUTE OFFSET IN DATA AND WEIGHTINGVRP(J) = VRP(J) + INPUT_ARRAY(INPUT_POINTER -INDEX) * 1 WEIGHTING(INDEX+1)20 CONTINUE 30 CONTINUEC C ****************** COMPUTE VECTOR OF OUTPUTS ******************C ****************** USING INVERSE FFT ROUTINE ****************** CCALL IFFT(VRP,OUTPUT_VECTOR,N) CC ****************** REMOVE RESIDUAL CARRIER TERM *************** CC ------------ IF M NOT EQUAL TO N, REMOVE CARRIER HERE CC ********************* NORMAL RETURN *************************** CRETURN CEND Stylized FORTRAN Example of an FDM-TDM Transmultiplexer 

#### Questions & Answers

List and explain four factors of production
Vuyo Reply
capital labour entrepreneur natural resources
Thembi
What is supply
Ogodo Reply
when the supply decreases demand also decreases
Thembi
types of demand and the explanation
akin Reply
what is demand
akin Reply
other things remaining same if demend is increases supply is also decrease and if demend is decrease supply is also increases is called the demand
Mian
if the demand increase supply also increases
Thembi
you are wrong this is the law of demand and not the definition
Tarasum
Demand is the willingness of buy and ability to buy in a specific time period in specific place. Mian you are saying law of demand but not in proper way. you have to keep studying more. because its very basic things in Economics.
Hamza
Demand is the price of Quantity goods and services in which consumer's are willing and able to offer at a price in the market over a period of time
Umar
what is commercialization?
Doris Reply
How to talk loan for bank?
Alfred Reply
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Ritisha Reply
Answer: GPA stands for Grade Point Average. It is a standard way of measuring academic achievement in the U.S. Basically, it goes as follows: Each course is given a certain number of "units" or "credits", depending on the content of the course.
Yusuf
what is small and Microbuisenes
tadesse Reply
What is fiscal policy
Dansofo
Who is the funder of Economic
Dansofo
founder , that is Adam Smith
Daniel
what is model
Daniel Reply
The wealth of Nations
Yusuf Reply
the wealth of nations, is it the first?
Umar
Yes very sure it was released in 1759
Yusuf
thank you Yusuf.
Umar
then when did he died?
Umar
17 July 1790 Born: 16 June 1723, Kirkcaldy, United Kingdom Place of death: Panmure House, Edinburgh, United Kingdom
Yusuf
1790
Yusuf
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Umar
what is fiscal policy
kemigisha Reply
what's mode?
Umar Reply
mode is the highest occurring frequency in a distribution
Bola
mode is the most commonly occurring item in a set of data.
Umar
Please, what is the difference between monopoly and monopsony?
Olaleye Reply
is there monopsony word?
Umar
I have no idea though
Umar
please, in which year Adam smith was born?
Umar
monopsony is when there's only one buyer while monopoly is when there's only one producer.
Bola
who have idea on Banter
Ibrahim
like trade by barter?
Bola
Monopoly is when there's excessively one seller and there is no entry in the market while monopsony is when there is one buyer
kemigisha
Adam smith was born in 1723
Bola
(uncountable) Good humoured, playful, typically spontaneous conversation. verb (intransitive) To engage in banter or playful conversation. (intransitive) To play or do something amusing. (transitive) To tease mildly.
Umar
which book Adam smith published first? the first book of Adam smith pls.
Umar
wealth on nation, 1776
Daniel
what is market power and how can it affect an economy?
Gab Reply
market power:- where a firm is said to be a price setter.market power benefits the powerful at the expense of others.
Umar
Market power refers to the ability of a firm (or group of firms) to raise and maintain price above the level that would prevail under competition is referred to as market or monopoly power. The exercise of market power leads to reduced output and loss of economic welfare
Kartheek
find information about the national budget
Molahlegi
three branches of economics in which tourism is likely to figure
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What are those three branches?
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Source:  OpenStax, An introduction to the fdm-tdm digital transmultiplexer. OpenStax CNX. Nov 16, 2010 Download for free at http://cnx.org/content/col11165/1.2
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