082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

798 Chapter 24 The Fourier transform
Engineeringapplication24.2
Effectoftruncatingthed.c.t.andd.f.t.ofasetofsamples
Now we turn our attention to a longer sequence and consider the effect of deleting
samples inthe d.c.t.
In this example, we produce both a d.c.t. and a d.f.t. of a given set of samples
andthensetsomeofthesamplesinthelatterpartofthesequencetozero.Following
this we carry out an inverse d.c.t. to return an approximation to the original sample
set.
We consider the sequence of samples
f[n]=0,1,2,3,4,5,6,7,8,9
Usingthe methods described inthe previous example wecan show that the d.c.t. is
F DCT
[k] = 14.2302,−6.3815,0.0000,−0.6835,0.0000,
−0.2236,0.0000,−0.0904,0.0000,−0.0254
ThenotationF DCT
hasbeenusedtodistinguishthissequencefromthed.f.t.sequence
thatitwill be compared to. The d.f.t. is
F DFT
[k] = 45.0000,−5.0000 +j15.3884,−5.0000 +j6.8819,−5.0000
+j3.6327,−5.0000 +j1.6246,−5.0000,−5.0000 −j1.6246,
−5.0000 −j3.6327,−5.0000 −j6.8819,−5.0000 −j15.3884
Bydefinition,applyingtheappropriateinverseoperationstoeithersetofdatawould
recover the originalinput samples.
If instead we delete samples from the end of each sequence by setting the values
to zero and then perform the inversion then the results are different. In this example
weset the lastfive samples tobezero. So weperform the inversion on
F DCT
[k] = 14.2302,−6.3815,0.0000,−0.6835,0.0000,0,0,0,0,0
F DFT
[k] = 45.0000,−5.0000 +j15.3884,−5.0000 +j6.8819,−5.0000
+j3.63271,−5.0000 +j1.6246,0,0,0,0,0
The results of the inversion are shown plotted in Figure 24.20. The values obtained
from the inverse d.c.t. are all real values. The values plotted on the graph for the
inversed.f.t.arethemagnitudesofthecomplexvaluesthatarereturned.Thisisnecessary
because now samples have been removed from the d.f.t., it can no longer be
guaranteed thatthe solution consists of wholly real numbers.
Notice that the samples in both cases are not exactly the same as the ones we
startedwith.Thisisexpectedbecauseinsettingsomeofthesamplevaluestozeroin
thed.c.t.wehavedestroyedsomedata.Infactwehavedestroyedhalfofthesamples
in this example. However, notice also that the d.c.t. appears to perform much better
than the d.f.t. in recovering the input data. The reason the d.c.t. looks qualitatively
better than the d.f.t. is that the information content or energy is concentrated in the
lowest order samples. Hence when we set the later samples to zero they affect the
d.c.t. far lessthan the d.f.t.