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1 Fourier Transform and Sampling Theorem Introduction The Fourier transform renders a mathematical approach that transforms a signal into its constituent frequencies. The original signal varies with time, and therefore it is the time domain representation of the signal, whereas the Fourier transform of the signal distributes with a range of frequency and is expressed in the frequency domain. The term Fourier transform refers both to the frequency domain representation of the signal and the mathematical approach that transforms the signal to its frequency domain representation. In this chapter, the definition and properties of Fourier transform will be given in Section 2.1. In Section 2.2, the application of Fourier transform will be explicated. In Section 2.3, Fourier Transform of periodic signals will be introduced. Principles of sampling theorem will be discussed in Section 2.4. 2.1 Definition of Fourier Transform Before Fourier Transform is applied, certain conditions must be satisfied. Dirichlet Condition: 1. () is absolutely able to be integral x(t) ! dt !! !! < . 2. The number of maxima and minima of () isfinite. 3. The number of discontinuity of () isfinite. Fourier Transform equation sets are as follow: = !!!!"# !! !! (2.1) = !!!"# !! !! (2.2) Some examples will be shown below to apply Fourier Transform. Example 2.1 = (), in time domain, see figure 2.1, find its Fourier Transform, and plot its spectrum in frequency domain. = !!!!"# = 1 !! !! Inversely: = 1 = () The spectrum plotted in frequency domain is shown in figure 2.2. Note: ” represents Fourier Transform Note: based on equation 2.2,

Fourier Transforms

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    Fourier Transform and Sampling Theorem Introduction The Fourier transform renders a mathematical approach that transforms a signal into its constituent frequencies. The original signal varies with time, and therefore it is the time domain representation of the signal, whereas the Fourier transform of the signal distributes with a range of frequency and is expressed in the frequency domain. The term Fourier transform refers both to the frequency domain representation of the signal and the mathematical approach that transforms the signal to its frequency domain representation. In this chapter, the definition and properties of Fourier transform will be given in Section 2.1. In Section 2.2, the application of Fourier transform will be explicated. In Section 2.3, Fourier Transform of periodic signals will be introduced. Principles of sampling theorem will be discussed in Section 2.4. 2.1 Definition of Fourier Transform Before Fourier Transform is applied, certain conditions must be satisfied. Dirichlet Condition:

    1. () is absolutely able to be integral x(t) !dt!!!! < . 2. The number of maxima and minima of () isfinite. 3. The number of discontinuity of () isfinite.

    Fourier Transform equation sets are as follow:

    = !!!!"#!!!! (2.1) = !!!"#!!!! (2.2)

    Some examples will be shown below to apply Fourier Transform. Example 2.1 = (), in time domain, see figure 2.1, find its Fourier Transform, and plot its spectrum in frequency domain.

    = !!!!"# = 1!!

    !!

    Inversely: = 1 = ()

    The spectrum plotted in frequency domain is shown in figure 2.2. Note: represents Fourier Transform Note: based on equation 2.2,

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    = !!!"#!!

    !!

    If = 1, = (), and = !!!"#!!!! .

    Figure 2.1 in time domain.

    Figure 2.2 in frequency domain.

    Example 2.2 = ( ), find its Fourier Transform.

    = !!!!"# = !!!!"#!!

    !!

    !!!!"# Note: The Fourier Transform of is:

    = !!!!"(!!!)!!

    !!

    Example 2.3 has its Fourier Transform , and ! = ( ), find the Fourier Transform of ! with respect to .

    = !!!!"#!!

    !!

    = ()!!!! !!!!"(!!!)

    = !!!!"# ()!!!! !!!!"#

    = !!!!"#

    Exercise: show that the Fourier Transform of !!!!!!

    ! is !!!

    .

    ( )t

    t

    ( )x f

    f

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    Example 2.4 has its Fourier Transform , and ! =!"(!)!"

    , find the Fourier Transform of ! .

    =

    ()

    !!!"#!!

    !!

    = 2()!!!!"!!!! = ! !!!"#

    !!!!

    Thus, the Fourier Transform of ! =

    !"(!)!"

    is 2! . Example 2.5 has its Fourier Transform , and ! = ()cos (2!), find the Fourier Transform of ! .

    ! = ! !!!!"#!!

    !!

    = () (2!)!!!!"#!!!!

    Note: cos 2! =

    !!(!!!!!! + !!!!!!!). Now, substitute it into equation above.

    ! =12 ()

    !!!! !!!! !!!

    !!+12 ()

    !!!! !!!! !!!

    !!

    = !! ! +

    !! + !

    Example 2.6 1 has its Fourier Transform , and = cos (2!), find the Fourier Transform of ! = 1 cos (2!).

    ! = 1 cos (2!)!!!!"#!!

    !!

    = !! ! +

    !! + !

    Note that the Fourier Transform of () is ().

    Example 2.7 = !!=

    !!

    !!! !

    !0

    , find the Fourier Transform of .

    =1

    !!!!"#!!!

    !!!

    = =

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    Question: What is the half power bandwidth of this spectrum? Answer: The half power bandwidth is calculated by selecting the magnitude drop to !

    ! (see

    figure 2.3), i.e., !"#$%&!"#

    = !! ,

    Using a numerical method it can be shown that the half power bandwidth is in the order of 1/T. In addition, from Figure 2.3, the zero-to-zero bandwidth is exactly equal to 2 / T.

    Figure 2.3 Fourier Transform of rectangular function

    Example 2.8 = cos (2!)

    !!

    , find the Fourier Transform of and plot its spectrum in frequency domain, see figure 2.4.

    =12 ! +

    12 + !

    = !! ! +

    !! + !

    Figure 2.4 in frequency domain.

    Example 2.9 Assume two digital sources are transmitting: = cos (2!)

    !!

    and

    = cos (2!) !!

    , find the Fourier Transform of = + () and plot its spectrum in frequency domain, see figure 2.1.5.

    = + ( )!!!!"#!!

    !!

    = ()!!!!"#!!!! + ()!!!!"#!!!!

    = !! ! +

    !! + !

    + !! ! +

    !! + !

    12

    sin ( )c fT

    f

    ( )X f

    f

    0f0f

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    Figure 2.5 Fourier Transform of

    Thus, unless ! and ! are far from each other, they may overlap. This overlap leads to interference effects. Note: By carefully selecting ! and !, we can maintain the frequency components orthogonal. This is the basis of DS-CDMA and MC-CDMA systems. See Example 2.10. Example 2.10 Calculate the time correlation of two signals cos 2! and cos 2! within the duration T. Here, T in digital communication represents the symbol or bit duration.

    cos 2! , cos 2! =1 cos (2! )cos (2!)

    !

    ! =

    24 +

    2(!+!)2(!+!)

    When = (!!) = , is an integer, and (!+!)T 1, first and second term becomes zero, respectively, and the correlation would be equal to 0, maintaining the orthogonality. Thus, orthogonality is maintained when:

    = / In practice, k = 1 is selected to maintain the spectrum usage minimal. Moreover, we should have:

    (!+!) 1/ or 2! 1/ From Example 2.7, recall that 1/ represents the half power bandwidth; thus, 2! 1/ refers to the fact that the carrier frequency should be much larger than the bandwidth, which refers to the fact that the carrier frequency should be much larger than the bandwidth. This type of signals are called narrowband signals. 2.2 Application of Fourier Transform 2.2.1 Modulation Schemes Example 2.11 DSB-SC Modulation, symbol means Fourier Transform, find the Fourier Transform of and plot its spectrum. See figure 2.6.

    = ()cos (!) () ()

    !!!!!! !

    ( )Z f

    f

    1f1f0f 0f

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    Note: !!!!!! !!!!"#!!!! = !!!!(!!!!)!!!!! = !

    The Fourier Transform of !!!!!! corresponds to:

    !!!!!! ! Thus, cos ! =

    !! [!!!!!! + !!!!!!!], then the Fourier Transform of is:

    =12 [ ! + + ! ]

    Figure 2.6 Fourier Transform of , spectrum of is shifted by to both sides

    2.2.2 Filtering When the signal is transmitted through the channel, the channel applies some filtering effects (see Figure 2.7).

    Figure 2.7 the output is the input convolved with filter The Fourier Transform of the convolution () is (). Now, if is applied to a system, the system would act as a filter. In Figure 2.8, it is assumed that ! > !, thus the output would be similar to the input.

    Figure 2.8 the output of () is well approximated by if the bandwidth of () is (much) higher than that of .

    Question: How about, if ! < ! ? Answer: If ! < !, some frequency component will be filtered out, in other word, the received signal cannot be fully recovered. See figure below.

    cf cf

    ( )X f ( )Y f

    h(t) ( )x t ( ) ( ) ( )y t x t h t=

    0f 0f1f1f =

    ( )X f ( )H f

    1f1f

    ( )X f

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    Figure 2.9 the output of () is well approximated by if the bandwidth of () is (much) higher than that of .

    2.3 Fourier Transform of periodic signals Using Fourier Transform series concept, any periodic signal can be represented by:

    = !!!! !!!

    !!!!!!! [! , ! + !] (2.3)

    Thus, the Fourier Transform of periodic signal can be represented by

    = !( !!!)!!!!!! (2.4)

    ! =!!!

    ()!!!!!!! !!!! !!!

    ! (2.5) Example 2.12: Consider the periodic signal = ( !)!!!!!! , ! = !/2. Periodic signal of :

    ! =1!

    ()!!/!

    !!!/!!!!!

    !!!

    ! =1!

    Fourier Transform of periodic signal of :

    =1!

    ( !

    )!!

    !!!!

    Example 2.13 Now consider the sampled signal that can be represented by:

    ! = !!!!!!! = ! !!!!!!! , Find its Fourier Transform. Given that the Fourier transform of () () is X(f).Y(f) ( stands for convolution), the Fourier transform of ! is:

    0f 0f1f1f =

    ( )X f ( )H f

    0f0f

    ( )X f

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    ! = 1!

    ( !

    )!!

    !!!!

    =1

    !(

    !

    )!!

    !!!!

    Note: this example is an important example for the study of communication system, because this example represents the principle of sampling theorem, see figure 2.10.

    Figure 2.10 the spectrum of

    From the figure 2.10, it manifests that if !

    !!= ! (sampling frequency) is bigger than2 (here,

    represents the highest frequency component of the bandlimited signal) then no overlap of the copies of the signal in frequency domain. In this case, the original signal can be recovered by applying a Low Pass Filter. If ! < 2, the replicas of the signal would overlap in frequency domain, which leads to Aliasing effect. The principle ! 2 is called the Nyquist Rate. 2.4 Sampling Theorem (Signal Reconstruction) As mentioned, applying a low pass filter (LPF) to the sampled signal, we can reproduce the originally sampled signal (see Figure 2.11).

    Figure 2.11: Reproducing the original signal.

    In Example 2.13, we show that ! =

    ! !!

    ( !!!)!!!!!! and ! = ! !!!!!!

    ! . Now, the goal is to apply a LPF, (), to ! to reconstruct . As we mentioned, low-pass filter can do it.

    = 12 <

    0

    ( )x t( )x t ( )X f

    0

    1T

    0

    1T

    Sampling LPF x(f) x(f)

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    = ( !

    !!) (2) ,

    = ! () =1

    !(

    !

    )!!

    !!!!

    ()

    The reconstructed signal is:

    = ! () = ! ! !!

    !!!!

    (2)

    = !

    !!

    !!!!

    (2 ! )

    Thus, if a band-limited signal is sampled at a rate higher or equal to 2, then the original signal can be reconstructed from the time sampled signal. However, if the sampling rate is less than 2, the original signal cannot be recovered and the Aliasing problem occurs. Accordingly, the Sampling Theorem is summarized as follow. Sampling Theorem: Let signal () be band limited with bandwidth W, i.e., let X(f) = 0 for |f| W. Let x(t) be sampled at multiples of some basic sampling interval Ts, be sampled at multiples of some basic interval Ts, where Ts 1/2W, to yield the sequence {x(nTs)} between n = to . Then, it is possible to reconstruct the original signal x(t) from the sampled values by the reconstruction formula.

    = x n! sinc[2W(t n!!! !! Ts)] where W is any arbitrary number that satisfies

    W W ( ! !!

    )W If ! =

    !!!

    (fs = 2W) the reconstruction relation simplifies to:

    = x n! sinc[(!!!) n!!! !! ] = x

    !!!

    sinc[2W(t!!! !!!!!

    )]