5.5 Discrete-time fourier transform (dtft)

Fundamentals of electrical Page 1 / 1

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 $1212$ . When the signal is real-valued, we can further simplify ourplotting chores by showing the spectrum only over $012$ ; 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 $12$ . 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 $12$ 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

12

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 $12$ and $12$ to unambiguously define it. When $a 0$ , we have a lowpass spectrum—the spectrum diminishes asfrequency increases from 0 to $12$ —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!

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.

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.

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:

1212 f 2 f m 2 f n 1 m n 0 m n δ m n

Therefore, we find that

f 1212 S 2 f 2 f n f 1212 m m s m 2 f m 2 f n m m s m f 1212 2 f m n s n

The Fourier transform pairs in discrete-time are

\begin{array}{l} S 2 f n \end{array}

∞ ∞ 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 1212 S 2 f 2 f 1212 n n s n 2 f n m m s n 2 f m, n m, n m s n s n f 1212 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

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?

Aislinn Reply

tijani

what is titration

John Reply

what is physics

Siyaka Reply

A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance

Jude Reply

Can you compute that for me. Ty

Jude

what is the dimension formula of energy?

David Reply

what is viscosity?

David

what is inorganic

emma Reply

what is chemistry

Youesf Reply

what is inorganic

emma

Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes

Adjei

please, I'm a physics student and I need help in physics

Adjanou

chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.

Pedro

A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity

Krampah Reply

2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.

Sahid Reply

you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s

Samuel Reply

can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?

Joseph Reply

Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time

Joseph

Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing

Joseph

"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true

Ryan

what's motion

Maurice Reply

what are the types of wave

Maurice

answer

Magreth

progressive wave

Magreth

hello friend how are you

Muhammad Reply

fine, how about you?

Mohammed

Mujahid

A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?

yasuo Reply

Who can show me the full solution in this problem?

Reofrir Reply

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!

100% Free Mobile Applications
Receive real-time job alerts and never miss the right job again

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

	15 Lec:15 Correlation Regression By Janet Forrester Start Quiz
©flickr: Jovial	Video Mid Term By Dewey Compton Start Exam
	2 Timeshare 2 By Jams Kalo Start Quiz
	2 Biology 02 The Chemical Foundation of Life MCQ By OpenStax Start Quiz
	5 Physiotherapy Flashcards Set 5 By Rhodes Start Flashcards
	20 Sociology 20 Population Urbanization Environment By OpenStax Start Quiz
©flickr: Abraham	2 Biology 2 By Sarah Warren Start Test
	Advertising Promotion BUS210 By Melinda Salzer Start Quiz
	11 Neuroanatomy 11 The Cerebellum By Stephen Voron Start Quiz
	SCJP Online Exam 310-065 By Prateek Ashtikar Start Quiz

5.5 Discrete-time fourier transform (dtft)

Spectrum of exponential signal

Spectra of exponential signals

Spectrum of length-ten pulse

Questions & Answers

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!