1. Home
  2. Fundamentals of electrical
  3. Digital signal processing
  4. Discrete-time fourier transform

5.5 Discrete-time fourier transform (dtft)

<< Chapter < Page
  Fundamentals of electrical   Page 1 / 1
Chapter >> Page >
Discussion of Discrete-time Fourier Transforms. Topics include comparison with analog transforms and discussion of Parseval's theorem.

The Fourier transform of the discrete-time signal s n is defined to be

S 2 f n ∞ ∞ s n 2 f n
Frequency here has no units. As should be expected, thisdefinition is linear, with the transform of a sum of signals equaling the sum of their transforms. Real-valued signals haveconjugate-symmetric spectra: S 2 f S j 2 f .

A special property of the discrete-time Fourier transform isthat it is periodic with period one: S 2 f 1 S 2 f . Derive this property from the definition of the DTFT.

S 2 f 1 n ∞ ∞ s n 2 f 1 n n ∞ ∞ 2 n s n 2 f n n ∞ ∞ s n 2 f n S 2 f
Got questions? Get instant answers now!

Because of this periodicity, we need only plot the spectrum overone period to understand completely the spectrum's structure; typically, we plot the spectrum over the frequency range 1 2 1 2 . When the signal is real-valued, we can further simplify ourplotting chores by showing the spectrum only over 0 1 2 ; the spectrum at negative frequencies can be derived frompositive-frequency spectral values.

When we obtain the discrete-time signal via sampling an analog signal, the Nyquist frequency corresponds to the discrete-time frequency 1 2 . To show this, note that a sinusoid having a frequency equal to the Nyquist frequency 1 2 T s has a sampled waveform that equals 2 1 2 T s n T s n 1 n The exponential in the DTFT at frequency 1 2 equals 2 n 2 n 1 n , meaning that discrete-time frequency equals analog frequency multiplied by the sampling interval

f D f A T s
f D and f A represent discrete-time and analog frequency variables, respectively. The aliasing figure provides another way of deriving this result. As the duration of eachpulse in the periodic sampling signal p T s t narrows, the amplitudes of the signal's spectral repetitions, which are governed by the Fourier series coefficients of p T s t , become increasingly equal. Examination of the periodic pulse signal reveals that as Δ decreases, the value of c 0 , the largest Fourier coefficient, decreases to zero: c 0 A Δ T s . Thus, to maintain a mathematically viable Sampling Theorem, theamplitude A must increase as 1 Δ , becoming infinitely large as the pulse duration decreases. Practical systems use a small value of Δ , say 0.1 · T s and use amplifiers to rescale the signal. Thus, the sampledsignal's spectrum becomes periodic with period 1 T s . Thus, the Nyquist frequency 1 2 T s corresponds to the frequency 1 2 .

Let's compute the discrete-time Fourier transform of the exponentially decaying sequence s n a n u n , where u n is the unit-step sequence. Simply plugging the signal'sexpression into the Fourier transform formula,

S 2 f n ∞ ∞ a n u n 2 f n n ∞ 0 a 2 f n

This sum is a special case of the geometric series .

n 0 ∞ α n α α 1 1 1 α

Thus, as long as a 1 , we have our Fourier transform.

S 2 f 1 1 a 2 f

Using Euler's relation, we can express the magnitude and phase of this spectrum.

S 2 f 1 1 a 2 f 2 a 2 2 f 2
S 2 f a 2 f 1 a 2 f

No matter what value of a we choose, the above formulae clearly demonstrate the periodicnature of the spectra of discrete-time signals. [link] shows indeed that the spectrum is a periodic function. We need only consider the spectrumbetween 1 2 and 1 2 to unambiguously define it. When a 0 , we have a lowpass spectrum—the spectrum diminishes asfrequency increases from 0 to 1 2 —with increasing a leading to a greater low frequency content; for a 0 , we have a highpass spectrum( [link] ).

Got questions? Get instant answers now!

Spectrum of exponential signal

The spectrum of the exponential signal ( a 0.5 ) is shown over the frequency range [-2, 2], clearly demonstrating the periodicity of all discrete-time spectra. The angle has unitsof degrees.

Spectra of exponential signals

The spectra of several exponential signals are shown. What is the apparent relationship between the spectra for a 0.5 and a 0.5 ?

Analogous to the analog pulse signal, let's find the spectrum of the length- N pulse sequence.

s n 1 0 n N 1 0

The Fourier transform of this sequence has the form of a truncated geometric series.

S 2 f n 0 N 1 2 f n

For the so-called finite geometric series, we know that

n n 0 N n 0 1 α n α n 0 1 α N 1 α
for all values of α.

Got questions? Get instant answers now!

Derive this formula for the finite geometric series sum. The "trick" is to consider the difference between theseries' sum and the sum of the series multiplied by α .

α n n 0 N n 0 1 α n n n 0 N n 0 1 α n α N n 0 α n 0 which, after manipulation, yields the geometric sum formula.

Got questions? Get instant answers now!

Applying this result yields ( [link] .)

S 2 f 1 2 f N 1 2 f f N 1 f N f
The ratio of sine functions has the generic form of N x x , which is known as the discrete-time sinc function dsinc x . Thus, our transform can be concisely expressed as S 2 f f N 1 dsinc f . The discrete-time pulse's spectrum contains many ripples, the number of which increase with N , the pulse's duration.

Spectrum of length-ten pulse

The spectrum of a length-ten pulse is shown. Can you explain the rather complicated appearance of the phase?

The inverse discrete-time Fourier transform is easily derived from the following relationship:

1 2 1 2 f 2 f m 2 f n 1 m n 0 m n δ m n
Therefore, we find that
f 1 2 1 2 S 2 f 2 f n f 1 2 1 2 m m s m 2 f m 2 f n m m s m f 1 2 1 2 2 f m n s n
The Fourier transform pairs in discrete-time are
S 2 f n ∞ ∞ s n 2 f n s n f 1 2 1 2 S 2 f 2 f n

The properties of the discrete-time Fourier transform mirror those of the analog Fourier transform. The DTFT properties table shows similarities and differences. One important common property is Parseval's Theorem.

n ∞ ∞ s n 2 f 1 2 1 2 S 2 f 2
To show this important property, we simply substitute theFourier transform expression into the frequency-domain expression for power.
f 1 2 1 2 S 2 f 2 f 1 2 1 2 n n s n 2 f n m m s n 2 f m , n m , n m s n s n f 1 2 1 2 2 f m n
Using the orthogonality relation , the integral equals δ m n , where δ n is the unit sample . Thus, the double sum collapses into a single sum because nonzero values occur only when n m , giving Parseval's Theorem as a result. We term n n s n 2 the energy in the discrete-time signal s n in spite of the fact that discrete-time signals don't consume(or produce for that matter) energy. This terminology is a carry-over from the analog world.

Suppose we obtained our discrete-time signal from values ofthe product s t p T s t , where the duration of the component pulses in p T s t is Δ . How is the discrete-time signal energy related to the total energycontained in s t ? Assume the signal is bandlimited and that the sampling ratewas chosen appropriate to the Sampling Theorem's conditions.

If the sampling frequency exceeds the Nyquist frequency, thespectrum of the samples equals the analog spectrum, but overthe normalized analog frequency f T . Thus, the energy in the sampled signal equals the original signal's energy multiplied by T .

Got questions? Get instant answers now!

Questions & Answers

Three charges q_{1}=+3\mu C, q_{2}=+6\mu C and q_{3}=+8\mu C are located at (2,0)m (0,0)m and (0,3) coordinates respectively. Find the magnitude and direction acted upon q_{2} by the two other charges.Draw the correct graphical illustration of the problem above showing the direction of all forces.
Kate Reply
To solve this problem, we need to first find the net force acting on charge q_{2}. The magnitude of the force exerted by q_{1} on q_{2} is given by F=\frac{kq_{1}q_{2}}{r^{2}} where k is the Coulomb constant, q_{1} and q_{2} are the charges of the particles, and r is the distance between them.
Muhammed
What is the direction and net electric force on q_{1}= 5µC located at (0,4)r due to charges q_{2}=7mu located at (0,0)m and q_{3}=3\mu C located at (4,0)m?
Kate Reply
what is the change in momentum of a body?
Eunice Reply
what is a capacitor?
Raymond Reply
Capacitor is a separation of opposite charges using an insulator of very small dimension between them. Capacitor is used for allowing an AC (alternating current) to pass while a DC (direct current) is blocked.
Gautam
A motor travelling at 72km/m on sighting a stop sign applying the breaks such that under constant deaccelerate in the meters of 50 metres what is the magnitude of the accelerate
Maria Reply
please solve
Sharon
8m/s²
Aishat
What is Thermodynamics
Muordit
velocity can be 72 km/h in question. 72 km/h=20 m/s, v^2=2.a.x , 20^2=2.a.50, a=4 m/s^2.
Mehmet
A boat travels due east at a speed of 40meter per seconds across a river flowing due south at 30meter per seconds. what is the resultant speed of the boat
Saheed Reply
50 m/s due south east
Someone
which has a higher temperature, 1cup of boiling water or 1teapot of boiling water which can transfer more heat 1cup of boiling water or 1 teapot of boiling water explain your . answer
Ramon Reply
I believe temperature being an intensive property does not change for any amount of boiling water whereas heat being an extensive property changes with amount/size of the system.
Someone
Scratch that
Someone
temperature for any amount of water to boil at ntp is 100⁰C (it is a state function and and intensive property) and it depends both will give same amount of heat because the surface available for heat transfer is greater in case of the kettle as well as the heat stored in it but if you talk.....
Someone
about the amount of heat stored in the system then in that case since the mass of water in the kettle is greater so more energy is required to raise the temperature b/c more molecules of water are present in the kettle
Someone
definitely of physics
Haryormhidey Reply
how many start and codon
Esrael Reply
what is field
Felix Reply
physics, biology and chemistry this is my Field
ALIYU
field is a region of space under the influence of some physical properties
Collete
what is ogarnic chemistry
WISDOM Reply
determine the slope giving that 3y+ 2x-14=0
WISDOM
Another formula for Acceleration
Belty Reply
a=v/t. a=f/m a
IHUMA
innocent
Adah
pratica A on solution of hydro chloric acid,B is a solution containing 0.5000 mole ofsodium chlorid per dm³,put A in the burret and titrate 20.00 or 25.00cm³ portion of B using melting orange as the indicator. record the deside of your burret tabulate the burret reading and calculate the average volume of acid used?
Nassze Reply
how do lnternal energy measures
Esrael
Two bodies attract each other electrically. Do they both have to be charged? Answer the same question if the bodies repel one another.
JALLAH Reply
No. According to Isac Newtons law. this two bodies maybe you and the wall beside you. Attracting depends on the mass och each body and distance between them.
Dlovan
Are you really asking if two bodies have to be charged to be influenced by Coulombs Law?
Robert
like charges repel while unlike charges atttact
Raymond
What is specific heat capacity
Destiny Reply
Specific heat capacity is a measure of the amount of energy required to raise the temperature of a substance by one degree Celsius (or Kelvin). It is measured in Joules per kilogram per degree Celsius (J/kg°C).
AI-Robot
specific heat capacity is the amount of energy needed to raise the temperature of a substance by one degree Celsius or kelvin
ROKEEB
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
<< Chapter < Page Page > Chapter >>

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Fundamentals of electrical engineering i. OpenStax CNX. Aug 06, 2008 Download for free at http://legacy.cnx.org/content/col10040/1.9
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Fundamentals of electrical engineering i' conversation and receive update notifications?

Ask