<< Chapter < Page Chapter >> Page >

The computational efficiency of the transmultiplexer can therefore be traced to two key items:

  1. Separation of the tuning computation into two segments, one of which (the { v ( r , p ) } ) need be computed only once
  2. The use of the FFT algorithm to compute the inverse DFT

The first accrues from strategic choices of the sampling and tuning frequencies, while the second depends on N being chosen to be a highly composite integer.

Figure four is a traditional cartesian graph in the first quadrant, with horizontal axis labeled Channels to be Tuned Simultaneously C, and vertical axis labeled Mega Multiply Adds. The values on the horizontal axis range from 0 to 40 in increments of 10, and the vertical axis ranges from 0 to 150 in increments of 50. There is one horizontal line labeled FFT-based Transmux with a vertical value of approximately 25. At (10, 25) and approximately (32, 25) there are small squares on the line. Beginning near the origin and increasing with a shallow, constant slope is a line labeled DFT-Based Transmux, with two hash-marks near the end of the graph. Beginning at the origin and increasing with a strong positive constant slope is a third line, labeled Conventional One-step Digital Tuning. There is a label in the middle of the figure that reads: System Parameters. f_s = 512 kHz, N = 128, Q = 16, ∆f = 4 kHz, M = N.
The Number of Multiply-Adds Needed to Compute C Tuner Outputs for a Particular Set of System Parameters
Figure 5 is a flow chart. Moving from left to right, the flow chart begins with the caption Analog FDM Signal. An arrow points to the right at a large rectangle containing the label A/D. Below this rectangle is an arrow pointing up, labeled f_s = N ⋅ ∆f. A large arrow points to the right from the A/D rectangle at a second rectangle that contains the caption Preprocessor. Below this rectangle are three bullet points that read, block points, weight, computer v(r, p). A large arrow again points to the right at a third rectangle labeled N Point Inverse DFT. To the right of this are a series of small horizontal lines, an arrow pointing to the top-left corner of the figure, and a large arrow pointing to the right. This series of lines and arrows is labeled Commutator.
The Basic FDM-to-TDM Digital Transmultiplexer

The transmux as a dft-based filter bank

We have just developed an FDM-TDM transmultiplexer by first writing the equations for a single, decimated digital tuner. The equations for a bank of tuners come from then assuming that (1) they all use the same filter pulse response and (2) their center frequencies are all integer multiples of some basic frequency step. In this section, we develop an alternate view, which happens to yield the same equations. It produces a different set of insights, however, making its presentation worthwhile.

Using the dft as a filter bank

Instead of building a bank of tuners and then constraining their tuning frequencies to be regularly spaced, suppose we start with a structure known to provide equally-spaced spectral measurements and then manipulate it to obtain the desired performance.

Consider the structure shown in [link] . The sampled input signal x ( k ) enters a tapped delay line of length N . At every sampling instant, all N current and delayed samples are weighted by constant coefficients w ( i ) (where w ( i ) scales x ( k - i ) , for i between 0 and N - 1 ), and then applied to an inverse discrete Fourier transform Whether or not it is implemented with an FFT is irrelevant at this point. Also, we happen to use the inverse DFT to produce a result consistent with that found in the proceeding subsction, but the forward DFT could also be used. . The complete N-point DFT is computed for every value of k and produces N outputs. The output sample stream from the m-th bin of the DFT is denoted as X m ( k ) .

Figure six is a flow chart involving three rows of labeled shapes. The first row is labeled Input data, followed by an arrow pointing to the right labeled x(k) at a box containing the caption z^-1. To the right is another arrow pointing right labeled x(k-1) at a box containing the caption z^-1. To the right of this is a longer unlabeled arrow pointing to the right at a third box containing the caption z^-1. Above this box is a caption that reads x(k-N+1). In the middle of each arrow, and after the last box, there are four arrows pointing down at four circles containing a large x. From left to right, these circles have arrows pointing at them from the left side that are labeled w_0, w_1, w_2, and w_N-1. Each circle also has an arrow below it pointing down at a long rectangle containing the caption N-point DFT. Aligned with the arrows above are four more arrows below this rectangle that point at the expressions below them that read from left to right, x_0(k), x_1(k), x_2(k), x_N-1(k).
Processing Weighted, Delayed Signals with Discrete Fourier Transform

Since DFTs are often associated with spectrum analysis, it may seem counterintuitive to consider the output bins as time samples. It is strictly legal from an analytical point of view, however, since the DFT is merely an N-input, N-output, memoryless, linear transformation. Even so, the relationship of this scheme and digital spectrum analysis will be commented upon later. We continue by first examining the path from the input to a specific output bin, the m-th one, say. For every input sample x ( k ) there is an output sample X m ( k ) . By inspection we can write an equation relating the input and chosen output:

X m ( k ) = p = 0 N - 1 x ( k - p ) w ( p ) e j 2 π m p N

the m-th bin of an N-point DFT of the weighted, delayed data. We can look at this equation another way by defining w ¯ m ( p ) by the expression

w ¯ m ( p ) w ( p ) · e j 2 π n p N

and observing that [link] can be written as

X m ( k ) = p = 0 N - 1 x ( k - p ) · w ¯ m ( p ) .

From this equation it is clear X m ( k ) is the output of the FIR digital filter that has x ( k ) as its input and w ¯ m ( p ) as its pulse response. Since the pulse response does not depend on the time index k , the filtering is linear and shift-invariant. For such a filter we can compute its transfer function, using the expression

Questions & Answers

what Make things scarce
Soudani Reply
why do we studied economic
Soudani
for the stability and growth of any reign.
Fayaz
what is macro economics.?
Fayaz
Google it
pokemon
which part of economics is interesting..
Fayaz
how does economics define me
clement Reply
what are the factors that determines the demand and supply
sillah
what are the importance of Economics
Betty Reply
tell me something very important about economics..
Fayaz Reply
how may I solve arithmetic mean ,,,all example
Cee Reply
what is agriculture
Itoe Reply
simple method of understanding cost concept
Oludare Reply
what is inflation
Christiana Reply
Inflation is a general increase in price levels
Zuko
is the action of inflating something
Abdifatah
inflation is the persistent increase in general price level of goods and services in an economy over a considerable period of time .
Tetteh
inflation is the general increase of a commodity in a particular period of time.
Turay
inflation is a general increase in price levels of commodities
shehu
what are the types of inflation?
Ebrima
inflation is the period of persistent rise in the general level of the price of goods services over time
Emmanuel
we have creeping inflation, demand pull inflation ,cost push inflation, and galloping inflation .
Emmanuel
I s Nigeria in any kind of in inflation? If yes which kind?
Martins
please can someone help me with the principles of economic?
Jonathan
hi
ahmed
How are you
Jonathan
fine
ahmed
ok
Jonathan
and u
ahmed
so to day what do u have
ahmed
am fine
Jonathan
nothing bro
Jonathan
ok
ahmed
who.?
Fayaz
no discassion just question
Aliraza
how can a location of a firm create difference between producers
joy Reply
what is monetary policy
joy
hello
Abdifatah
is a monetary from policy that's authorized of country encharces
Abdifatah
What would you say about the the mobility of enterprise as a factor of production?
Cathryn Reply
how can I connect myself to this Ambrose platform
kanu Reply
I am good and you I am from sierra Leone and I am new her
kanu Reply
u are welcome bro, here is a good platform for you to be
Alie
That i know,thanks bro.
what the main definition of economic
Uhara Reply
the main definition is given by prof Lionel Robbins as a social science which studies human behavior between ends and scarce which have alternative uses
olajumoke
what covers macro economics.
Fayaz
Essay about Microsoft
Kwena Reply
what is economics
Julie Reply
what do you mean by means in economics
Julie
economic is the wealth of a country.
Moussa
monetary policy is refer to as being expansionary or contractionary.
Abdul
pls who can help me to explain money market and capital market
Au
money market is base on short term loan which is within one year period while capital market is long term loan more than one year...
Muhammad
money market is a market were short term loans are dealt with while capital market is a market were long term loans are traded
Ebrima
What is mean by monetory policy
Lovely
monetary polices are rules that control the rate of monetary exchange in an economic as a whole.
Ebrima
wealth of the nation
Uhara
While the American heart association suggests that meditation might be used in conjunction with more traditional treatments as a way to manage hypertension
Beverly Reply
in a comparison of the stages of meiosis to the stage of mitosis, which stages are unique to meiosis and which stages have the same event in botg meiosis and mitosis
Leah Reply
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get the best Algebra and trigonometry course in your pocket!





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
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'An introduction to the fdm-tdm digital transmultiplexer' conversation and receive update notifications?

Ask