Relationships between bandwidth, Fourier trans & digital modulations

In summary, the conversation discusses the relationship between bandwidth and Fourier transformation in W. Stallings's "Wireless Communications & Networks, 2nd edition". The author explains that the bandwidth is determined by the difference between the frequency of the last sine term and the first sine term in a Fourier series approximation of a square wave. However, in digital modulation techniques like ASK, FSK, and PSK, binary bits are encoded as a single sine wave through manipulation of its amplitude, frequency, or phase, eliminating the need for bandwidth. However, QAM, which uses ASK and PSK, still requires bandwidth due to the introduction of sidebands during modulation. The conversation ends with a request for more information on the relationship between bandwidth and
  • #1
Eus
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HiHo!

W. Stallings's "Wireless Communications & Networks, 2nd edition" explains the relationships between bandwidth and Fourier transformation by depicting a square wave. The square wave is approximated with a Fourier series having several sine terms. The bandwidth is then defined by the difference between the frequency of the last sine term and the first sine term (e.g., the bandwidth of [tex]\frac{4}{\pi}(sin((2\pi \times 10^{6})t) + \frac{1}{3}sin((2\pi \times 3 \times 10^{6})t) + \frac{1}{5}sin((2\pi \times 5 \times 10^{6})t))[/tex] is [tex](5 \times 10^{6}) - (1 \times 10^{6}) = (4 \times 10^{6})[/tex] Hz).

Okay, so I understand that the bandwidth is needed to properly approximate the square wave.

However, in the same chapter, the author explains about digital modulation techniques like ASK (Amplitude Shift Keying), FSK (Frequency Shift Keying) and PSK (Phase Shift Keying). I see that in the techniques, binary bits are not encoded as a square wave but as a single sine wave through the manipulation of its amplitude, frequency or phase.

So, I understand that since there is only a single sine wave, there is no bandwidth requirement anymore in ASK and PSK since they only use a single frequency (FSK has a bandwidth requirement since different frequencies are needed to encode different bits).

Is that true? Or, is it to naive? Any pointer to literature to understand the relationships better?

What confuses me is that why I still hear the word bandwidth when talking about QAM that only uses ASK and PSK?


Eus
 
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  • #2
I thought of the same thing before and I still don't have an answer to your last question, but I'd guess it might be because QAM is often combined with FDM like ADSL and total bandwidth goes way higher than original carrier wave ?
 
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  • #3
Well, in that case, I guess this is an expert question on a transceiver system. No wonder the book has a considerable gap between the FFT explanation and digital signal modulation.

Does anyone have a pointer to understand the matters within the gap?
 
  • #4
The only signal that requires "zero bandwidth" is a pure sine wave -- a wave that lasts indefinitely, and never changes in amplitude, phase, or frequency.

If you change any of those parameters during transmission, you no longer have a pure sine wave. Abrupt changes from one amplitude to another, for example, actually involve frequency components on both sides of the carrier. The spectral content introduced by modulation is usually called the "sidebands."

You can read more about sidebands on Wikipedia.

- Warren
 
  • #5
Hi Warren!

Thank you for the enlightening information. That covers the gap nicely :-)
 
  • #6
Yes, in QAM you do start with a single, reference signal, whose timing must be agreed by the terminating modems. However this nominally sinusoidal tone soon gets very rude things done to it ~ the constant chopping and changing of phase and amplitude leaves an oscillogram of visually incomprehensible squiggles.
If a Fourier analyser is attached to the output of a QAM modem, you will see that a wide frequency band is required to convey the QAM signal.
 

FAQ: Relationships between bandwidth, Fourier trans & digital modulations

What is the relationship between bandwidth and digital modulations?

The bandwidth of a digital modulation refers to the range of frequencies required to accurately transmit the digital signal. The higher the bandwidth, the more data can be transmitted at once. Digital modulations use a specific coding scheme to convert digital signals into analog waveforms that can be transmitted over a specific bandwidth.

How does Fourier transform relate to digital modulations?

Fourier transform is a mathematical tool used to analyze signals in the frequency domain. It is used to decompose a signal into its individual frequency components. In digital modulations, the signal is converted into a series of discrete frequencies and then transmitted over a specific bandwidth. Fourier transform is used in the design and analysis of digital modulations to ensure optimal use of the available bandwidth.

What is the impact of bandwidth on the quality of digital modulations?

The bandwidth has a direct impact on the quality of digital modulations. A larger bandwidth allows for more data to be transmitted, resulting in higher data rates and potentially better signal quality. However, using a larger bandwidth also increases the chances of interference from other signals, which can negatively affect the quality of the transmission.

How do bandwidth and Fourier transform affect the performance of digital modulations?

The relationship between bandwidth and Fourier transform is crucial in the performance of digital modulations. The bandwidth determines the range of frequencies available for the transmission, while Fourier transform ensures that the signal is properly encoded and decoded at the receiver end. The proper combination of bandwidth and Fourier transform parameters can result in efficient and high-quality digital modulations.

What are some common types of digital modulations and their corresponding bandwidth requirements?

Some common types of digital modulations include Amplitude Shift Keying (ASK), Frequency Shift Keying (FSK), and Phase Shift Keying (PSK). Each of these modulations has different bandwidth requirements, with ASK requiring the least and PSK requiring the most. The bandwidth requirements also depend on factors such as the data rate, signal-to-noise ratio, and modulation scheme used.

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