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3.6.b Existence and properties of the DTFT
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Из курса от партнера École Polytechnique Fédérale de Lausanne
Digital Signal Processing
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Module 3: Part 2 - Advanced Fourier Analysis
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Paolo PrandoniLecturer
School of Computer and Communication Science
Martin VetterliProfessor
School of Computer and Communication Sciences
So existence means simply that the sum that defines a DTFT does not blow up.
This is easy to prove for absolutely summable sequences.
If you take the magnitude of the DTFT at any point omega,
this is equal to the sum for n that goes from minus infinity to
plus infinity of x[n] times e to the- j omega n in magnitude.
Now, every time you have the absolute value of a sum, you know that this is
maximized by the sum of the absolute values of the elements of the sum.
So we do this and because the magnitude of the complex exponential is one,
this is actually equal to the sum of the absolute values of the sequence.
Since our initial hypothesis was that the sequence was absolutely summable,
this is less than infinity, and therefore the DTFT exists for all values of omega.
Similarly, we can invert the DTFT very easily if we
assume absolute summability of the underlying sequence.
As we showed in the previous module,
the inversion formula is 1 over 2 pi times the integral between minus pi and
pi of the DTFT times e to the j omega n in d omega.
So we replace x of e to the j omega by the definition of
the DTFT in here, and because of the absolute summability of the sequence,
we can invert the summation and the integral.
When we do that, we have the sum for
k that goes from minus infinity to plus infinity of x[k] time this integral here.
So each element of the sequence in the sum is multiplied by this integral which
depends on k.
But now look at the numerator of this fraction here.
This is a complex exponential, and if n is different than k,
this will span an integer number of periods in the -pi, pi integral.
And therefore, the integral will be 0.
So what that means is that this integral is 0 unless n is equal to k,
at which point all the elements in the sum will be killed except for x[n].
And so in the end we have the result we're looking for.
The DTFT looks exactly like an inner product in the space C infinity.
If you take this inner product here between an infinite sequence and
the sequence e to the j omega n, and you write the definition of the inner product,
you get exactly the formulation for the DTFT.
The problem here is that C infinity is not really a well defined vector space.
There are sequences that do not converge in C infinity.
Nonetheless if we manage to establish a formal parallel between a change
of basis and the DTFT, it will mean that everything we discovered about
a DFT which is a well defined entity, will apply to the DTFT as well.
And all our intuition about the frequency domain will translate to the DTFT.
Now the basis here, that we're talking about, is not really a basis, because it's
an infinite and uncountable set of vectors indexed by a real valued variable omega.
So something breaks down really.
We start with sequences, but we end up landing in a space of functions.
On top of it all although we just proved existence and invertibility for
absolutely summable sequences, in reality the DTFT exists for
all square-summable sequences, which is a larger set of sequences.
But in that case the proofs we just gave become much more technical and
so we will skip them here.
Let's sum up the situation so far.
For finite length signals, we start in CN, and
by a change of basis, we compute their representation in the frequency domain,
which as well lives in CN.
We can go back to the original sequence via the inversion formula.
And the basis that allows us to go from the time domain to
the frequency domain is the DFT basis, the Fourier basis for
the DFT, which is a countable set of n Fourier basis vectors.
The DFS is exactly the same.
The expansion and reconstruction formulas are the same,
except that in this case we assume that everything is periodic underneath.
And now we have the DTFT we start from the space of a square summable sequences.
And via a formal change of basis, so, basis here is in quote, we end
up in the space of square integrable functions on the interval minus pi, pi.
By looking at the DTFT as a formal basis of expansion,
the linearity property follows easily from the linearity of the inner product.
So, the DTFT of a linear combination of two sequences
will be the linear combination of the DTFTs.
A second property that is easy to
prove from the definition of the DTFT is a time shift property.
So, if we take a sequence and we shift it in time by big M samples,
the DTFT of this shifted sequence is equal to the DTFT of
the original sequence times a delay factor.
e to the -j omega big M, which is very similar to what we obtained in
the case of the DFS when we took the shift of a periodic sequence.
The dual of this property is the modulation property of the DTFT.
So if we take a sequence and
we multiply this by a complex exponential at frequency omega 0,
what happens in frequency is that we have a shift of the spectrum by omega 0.
The time reversal property tells us that the Fourier transform of a time reversed
sequence, a sequence where we flip the values across the origin,
will be equal to a frequency reversed Fourier transform.
And the conjugation property says that if you conjugate every value of the sequence,
the Fourier transform will be both conjugated and frequency reversed.
Now, some particular cases that are very useful to remember,
because they appear often, first of all,
if the sequence is symmetric, then the DTFT is symmetric as well.
If the sequence is real, then the DTFT is Hermitian-symmetric.
The reality of the sequence can be expressed mathematically by
saying that x[n] is equal to the conjugate of x[n].
This infrequency implies that the Fourier transform of the sequence is equal to
the conjugate and frequency reversed version of the Fourier transform.
A simple corollary of this property is the fact that if x[n] is real,
then the magnitude of DTFT is symmetric.
You can verify this simply by taking the magnitude of both terms of this equation.
And an even more special case states that if x[n] is real and
symmetric, then X of j omega is also real and symmetric.
So this all looks nice and fine.
It looks like we have a full fledged basis expansion and
that the DTFT is just another version of the DFT.
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