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I read a digital communication problem and they interchange terms as windowing function and pulse shaping and refer to it as same entity. For example the following windowing function with Transition time $T_R$ used to smooth transitions between symbols is also refered to as pulse shaping

$w_T(t)= \left\{ \begin{array}{ll} \sin^2(\frac{\pi}{2}(0.5+\frac{t}{T_{TR}})) & \mbox{if $-T_{TR}/2 < t<T_{RT}/2$};\\ 1 & \mbox{if $T_{TR}/2 < t<T-T_{TR}/2$}\\ \sin^2(\frac{\pi}{2}(0.5-\frac{t-T}{T_{TR}})) & \mbox{if $T-T_{TR}/2 < t<T+T_{TR}/2$};\\ \end{array} \right. $

My understanding is that a pulse shaping function limits signal bandwidth to transmission bandwidth.

How come the two are refered to as same?

Thanks

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Windowing and pulse shaping are very similar, and are implemented in similar ways, but their purposes are different.

  • Windowing means multiplying a discrete signal by a pulse with certain spectral properties before calculating its Fourier transform. The purpose is reducing the effect of discontinuities at both ends of the discrete signal.

  • In QAM and related digital signals, the information is transmitted in the amplitudes of a sequence of pulses. Pulse shaping means choosing a pulse with the desired spectral and orthogonality properties.

Note that some digital communications textbooks assume (many times implicitly) a rectangular pulse shape, which is then converted to another pulse shape (using a filter, as in the example in your question). In my opinion, this formulation is unncecessarily complex -- it is better to explicity select a pulse shape from the start.

Note also that pulse shaping may be understood as the convolution of a stream of deltas with a pulse shape; in that sense, pulse shaping is a sort of "dual" of windowing.

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  • $\begingroup$ Thanks MBaz but I have question. So in your opinion does this windowing function I provided here also performs pulse shaping? $\endgroup$
    – Henry
    Commented Jun 14, 2015 at 16:42
  • $\begingroup$ Yes. It's a matter of context. You can multiply $w_T(t)$ by a sequence before calculating its FT (windowing). Or you can use it as a filter to shape (in time and frequency) a digital communications signal (pulse shaping). $\endgroup$
    – MBaz
    Commented Jun 14, 2015 at 16:52
  • $\begingroup$ Actually this function $W_T(t)$ in the application I saw is being used after the IFFT operation for an OFDM symbol before transmission over the bandwidth. $\endgroup$
    – Henry
    Commented Jun 14, 2015 at 16:56
  • $\begingroup$ It is used for pulse shaping, then. $\endgroup$
    – MBaz
    Commented Jun 14, 2015 at 17:08
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    $\begingroup$ @Henry In OFDM, the pulse shape of all subcarriers is implicitly rectangular due to the usage of the IDFT. Windowing of the IDFT output signal is usually applied to shape the spectrum of the transmit singal as a whole. The term "windowing" is used in order to not confuse it with the ususal concept of "the convolution of a stream of deltas with a pulse shape" as MBaz has nicely described pulse shaping. $\endgroup$
    – Deve
    Commented Jun 15, 2015 at 7:36

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