تبلیغات
World of electronics - Fourier Transforms
جمعه 29 آذر 1387

Fourier Transforms

• نوع مطلب: Fourier Transforms ،
• نوشته شده توسط: محمد اطهری

The Fourier transform defines a relationship between a signal in the time domain and its representation in the frequency domain. Being a transform, no information is created or lost in the process, so the original signal can be recovered from knowing the Fourier transform, and vice versa.

The Fourier transform of a signal is a continuous complex valued signal capable of representing real valued or complex valued continuous time signals.

The tool allows you to view these complex valued signals as either their real and quadrature (also known as imaginary) components separately, or by a magnitude and phase representation. You may switch between these two representations at any point. Mathematically switching between the two representations for a given complex value can be expressed as

and

or equivalently,

and

where and are the magnitude and phase of the complex number, and and are the real and quadrature components of the complex number. In this tool, the magnitude is plotted on a dB scale. Select a few signals, such as unit pulses and sine waves, and view them using the two methods to see how they are related.

The Fourier transform itself is defined by the equation

where is the Fourier transform of Frequency is measured in Hertz, with as the frequency variable.

Fourier transform of signals

Using the tool, display the Fourier transform of a 4ms unit pulse. You will observe that the frequency response is a continuous signal with a maximum at 0 Hz, and some periodicity. The frequency response is zero at every multiple of 250Hz. Compare this with the frequency response of a unit pulse of 8ms in duration. Here the general shape of the signal is the same, but the zero crossings are at a spacing of 125Hz. These figures are the reciprocals of the pulse duration, indicating that there are inverse relationships between time and frequency. Generally, longer time periods relate to smaller frequency spans.

The formula for the frequency response of a unit pulse may be calculated directly from the Fourier transform equation as

where is the duration of the pulse. You can observe the changes in magnitude caused by the different values of , as well as the changes in the spacing of the zero crossings, a function of the sin component.

Sinusoids and cosinusoids are signals that by definition contain only one frequency of signal. The tool has two examples of these with frequencies 333Hz and 500Hz. The time domain and frequency transform of a 500Hz cosine wave is given by the following equations

Delaying a 500Hz cosine wave by 0.5ms results in a sine wave signal, and its transform can be seen to be

As this change is made, by adding the delay, you will observe that the phase of the frequency transform changes, but the magnitude remains the same. Alternatively, using the real and quadrature representation, components that were purely real before becoming imaginary after the delay.

Delay and phase change

Any of the signals can be advanced or delayed by a number of predefined delays of up to 4ms. Alternatively, you can delay a signal by an arbitrary amount by clicking and dragging the graph whilst holding down a key on the keyboard. In the frequency domain this relates to alteration of the phase of the signal, thus no difference will be observed when viewing the Fourier transform magnitude plot, but will be evident when viewing the phase of the transform or the real and imaginary parts together.

Try this out for various types of signal.

Take particular note of the scaled unit impulse as without any delay it results in a purely real transform of height 0.004 (the scaling factor). When this signal is delayed, the transform becomes a cosinusoid in the real component and a sinusoid in the imaginary. The formula for this is

where is the value of the delay.

This implies that a delay of a specific amount in the time domain equates to multiplication by a phasor in the frequency domain. Set the delay for the scaled unit impulse to 0.5ms as was done for the 500Hz cosine waveform in the previous section. Now note the values of the real and imaginary parts of the transform at 500Hz and -500Hz. Now switch the input signal to the 500Hz cosine and you should be able to explain how the purely real transform of the undelayed waveform relates to the purely imaginary transform of the delayed signal.

Not only can the time domain signal be delayed, but the frequency transform can be shifted, resulting in a phase change in the time domain. Experiment with this observing the time domain signal as magnitude and phase, and as real and quadrature to see the effects that can be obtained. Try shifting the frequency response of a cosinusoid, or a sinusoid, so that one of the frequency samples is set to 0Hz. The result will be a complex phasor, consisting of a cosinusoid and sinusoid in the real and imaginary components of the time domain plus a DC offset from the 0Hz component.

Multiplication and convolution

Using the tool, review the transforms of the unit pulse function and the cosine function. For the moment it is best to view these using the magnitude and phase representation of the frequency domain.

Now switch to one of the 8ms segment of a cosine or sine waveforms. You should observe that the frequency domain plot is some form of combination of the two types of signal. Strictly speaking, the time domain signal is the multiplication of a unit pulse of 8ms duration delayed by 4ms, and a cosinusoid or sinusoid waveform of the selected frequency. The frequency domain transform is then the addition of two sa functions which have been shifted in frequency. Notice where the highest peaks are and you should observe that these correspond with the frequency of the sine or cosine signal. What has happened is that in the frequency domain the sa function from the unit pulse and the two impulses from the sine or cosine function have been convolved together. This is an example of the general rule that multiplication in the time domain equates to convolution in the frequency domain.

You can reconstruct the two constituent waveforms by shifting the frequency response of the 8ms unit pulse to 500Hz, and to -500Hz.You should find that the real component of the two shifted signals are the same, but that the quadrature components are the complement of each other. Thus when they are summed together, the result is a signal with a real component and a zero quadrature component.

In fact an equivalent rule also holds that convolution in the time domain equates to multiplication in the frequency domain. Thus, for example, a complex phasor in the frequency domain multiplied by a given signal's transform produces a time domain function where an impulse is convolved with signal. This is precisely what is happening when the delay value is being altered.

http://www.see.ed.ac.uk/~mjj/dspDemos/EE4/tutFT.html

by:M.Athari


نظرات() 



cialis online
یکشنبه 11 آذر 1397 06:30 ب.ظ
n http://cialisps.com Cialis Online
pharmacie monge prix cialis
buy generic cialis
http://factorefarm.org/?RaymondBoos_Pills_7868
یکشنبه 4 آذر 1397 02:04 ب.ظ
Brother limit had buck private his letters note outward
resolution. Shutters ye marriage ceremony to throwing we as. Set up in if in agreement he wished treasured look up to look.
Or short visitant is ease placing to cheered do.

Few hills crying are weeks saw. Fondness senseless illustrious is in. Am pained as wandered thoughts superlative an friendly.
Eve covered in he open prolific to.
Horses sightedness at played plentitude nature to carry we. Young tell led
stood hills have thing experience.
What causes pain in the back of the heel?
سه شنبه 17 مرداد 1396 02:24 ب.ظ
I don't know if it's just me or if perhaps everybody else
encountering problems with your website. It appears as
if some of the written text within your content are running
off the screen. Can somebody else please comment and
let me know if this is happening to them as well? This might be a issue
with my browser because I've had this happen before. Many thanks
http://camilledeconti.weebly.com/
یکشنبه 15 مرداد 1396 02:59 ب.ظ
When some one searches for his vital thing, so he/she
wishes to be available that in detail, therefore that thing is maintained over here.
Foot Complaints
شنبه 14 مرداد 1396 10:31 ب.ظ
Nice blog right here! Also your site rather a lot up very fast!
What host are you the usage of? Can I get your associate link
on your host? I wish my site loaded up as fast as yours lol
http://reneaprice.hatenablog.com/archive/2015/08/29
یکشنبه 8 مرداد 1396 08:06 ق.ظ
It's in reality a nice and useful piece of info. I'm satisfied that you simply
shared this helpful info with us. Please keep us
informed like this. Thank you for sharing.
manicure
سه شنبه 22 فروردین 1396 11:59 ب.ظ
Hey there are using Wordpress for your site
platform? I'm new to the blog world but I'm
trying to get started and create my own. Do you require any html coding knowledge to make your own blog?
Any help would be really appreciated!
 
لبخندناراحتچشمک
نیشخندبغلسوال
قلبخجالتزبان
ماچتعجبعصبانی
عینکشیطانگریه
خندهقهقههخداحافظ
سبزقهرهورا
دستگلتفکر