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some of the properties of Fourier tansform and indicate how those properties are used for a variety of things.
a synthesis equation(x[n]) and an analysis equation(x(Ω)).
synthesis equation essentially corresponds to decomposing the sequence as a linear combination of complex exponentials with amplitudes that are in fact proportional to the Fourier transform
The Fourier tansform is a periodic function of frequency.
One is the time shifting property, and the time shifting property tells us that if X(Ω) is the Fourier transform of x[n] then the Fourier transofrm of x[n] shifted in time is that the same Fourier transform multiplied by this factor which is a liner phase
factor. So a time shifting introduces a linear phase term. And by the way recall that in the continuous time case we had a similar situation namely that a time shift corresponded to a linear phase. There also is a dual to the time shifting property which is
referred to as the frequency shifting property which tells us that if we multiply a time function by a complex exponential that in fact generates a frequency shift and we will see this frequency shifting property surface in a slightly different way shortly
when we talk about the modulation property in the discrete time case. Another important property we will want to make use of shortly is linearity which follows in a very straightforward way from the Fourier transform definition. And the linearity property
says simply that Fourier tansform of a sum or linear combination is the same linear combination of the Fourier transforms. The convolution property is the property that tells us how to relate the Fourier transform of the convolution of 2 sequences to the Fourier
transforms of individual sequences. What happens and this can be demonstrated algebraically. The Fourier transform of the convolution is simply the product of the Fourier transforms. So Fourier transform maps convolution in the time domain to multiplication
in the frequency domain.
Specifically the argument was that the Fourier transform of a sequence or a signal corresponds to decomposing it into a linear combination of complex exponentials.
a first order difference equation is a filter, and in fact it's a very important class of filter and it's used very often to do approximate low pass and high pass filtering.
Now in addition to the convolution property another important property that we had in the continuous time and we have in the discrete time is the modulation property. The modulation property tells us what happens in the frequency domain when you multiply
signals in the time domain. In continuous time, the modulation property corresponded to the statement that if we multiply in the time domain, we convolve the Fourier transforms in the frequency domain. And in discrete time, we have very much the same kind
of relationship. The only real distinction between these is that in the discrete time case in carrying out the convolution, it's an integration only over a 2π interval. And what that corresponds to is what's referred to as a periodic convolution. As opposed
to the continuous time case, what we have is a convolution that is an aperiodic convolution. So again we have a convolution property in discrete time that is very much like the convolution property in continuous time. The only real difference is that here
we are convolving periodic functions and so it's a periodic convolution which involves an integration only over 2π interval rather than an integration from minus infinity to plus infinity.
Let's take a look at an example of the modulation property which will then lead to one particular application and a very useful application of the modulation property in discrete time. The example that I want to pick is an example in which we consider modulating
a signal with another signal x[n], essentially what that says is that any signal which I modulate with this in fact corresponds to taking the original signal and then going through that signal alternating the algebraic signs. Now in applying the modulation
property of course what we need to do is develop the Fourier transform of the signal. This signal which I can write it as (-1)^n is a periodic signal. And it's the periodic signal I show here. And recall that to get the Fourier transform of a periodic signal
one way to do it is to generate the Fourier series coefficients for the periodic signal and then identify the Fourier transform as an impulse chain where the heights of the impulses in the impulse chain are proportional with a proportionality factor of 2π.
If we have a system which corresponds to a low pass filter as I indicate here. If we want to convert that to a high pass filter, we can do that by generating a new system whose impulse response is (-1)^n times the impulse response of the low pass filter,
and this modulation by (-1)^n will take the frequency response of this system and shift it by π, so that what's going on here at low frequencies will now go on here at high frequencies. This also says incidentally that if we look at an ideal low pass filter
and an ideal high pass filter and we choose the cut-off frequencies for comparison or the band width of the filter to be equal. Since this ideal high pass filter is this ideal low pass filter with the frequency response shifted by π.
The modulation property tells us that in the time domain what that corresponds to is an impulse response multiplied by (-1)^n. So ti says that the impulse response of the high pass filter or equivalently the inverse Fourier transform of the high pass filter
frequency response is (-1)^n times the impulse response of the low pass filter. that all follows the modulation property.
Now there is another way interesting and useful way that modulation can be used to implement or converse from low pass filtering to high pass filtering. The modulation property tells us about multtiplying in the time domain and shifting in the frequency
domain. The example that we happened to pick said if you multiply or modulate by (-1)^n, that takes low frequencies and shift some to high frequencies, what that tells us as a practical and useful notion is the following. Suppose we have a system that
we know is a low pass filter and it's a good low pass filter. How might we use it as a high pass filter. One way to do it instead of shifting its frequency response is to take the original signal shift its low frequencies to high frequencies by multiplying
the input signal the original signal by (-1)^n. Process that with a low pass filter where now with seating of low frequencies with high frequencies, and then unscramble it all at the output. So then we put the frequencies back where they belong. And I summarize
that here lets suppose for example that this system is a low pass filter. which with seating of low frequencies here with high frequencies of this signal. And then after the low pass filtering moving the frequencies back where they belong by again by modulating
with (-1)^n.
There is one more very important piece to the duality relationships. And we can see that first algebraically by comparing the continuous time Fourier series and the discrete time Fourier transform. The continuous time Fourier series in the time domain is
a periodical continuous function, in the frequency domain is an aperiodic sequence. In the discrete time case in the time domain, we have an aperiodic sequence and in the frequency domain, we have a function of a continuous variable which we know is periodic.
And so in fact we have in the time domain here aperiodic sequence, in the frequency domain we have a continuous periodic function. Then what we see in fact is a duality between the continuous time Fourier series and the discrete time Fourier transform. But
the Fourier transform in discrete time is a periodic function of frequency. That periodic function has a Fourier series repressentation what is this Fourier series?
Also we talked about the Fourier transform both continuous time and discrete time, two important properties that we focused on among many of the properties are the convolution property and the modulation property. We have also shown that the convolution
property this very important concept namely filtering the modulation property leads to an important concept namely modulation. We have also very briefly indicated that how these properties and how these concepts have practical implications