Phase function is an odd function of ϖ ˆ making use

Info icon This preview shows pages 9–13. Sign up to view the full content.

View Full Document Right Arrow Icon
phase function is an odd function of ϖ ˆ Making use of the even property of the magnitude spectrum of real discrete-time sequence, the spectrum can sufficiently be plotted on the frequency range [0,π). (iv) Time shifting If the data sequence x [ n ] is shifted by an integer index of n 0 , the DTFT has a phase shift by an amount ϖ ˆ n 0 at frequency ϖ ˆ . ) ( ] [ ˆ ˆ 0 0 ϖ ϖ j n j e X e n n x - - Proof: -∞ = + - -∞ = - = - = - k n k j n n j e k x e n n x n n x F ) ( ˆ ˆ 0 0 0 ] [ ] [ ]} [ { ϖ ϖ ) ( ] [ ˆ ˆ ˆ ˆ 0 0 ϖ ϖ ϖ ϖ j n j k k j n j e X e e k x e - -∞ = - - = = (v) Time reversal Reversing the time sequence has the DTFT reversed in frequency. ) ( ] [ ˆ ϖ j e X n x - - Proof: -∞ = -∞ = - = - = - m m j n n j e m x e n x n x F ϖ ϖ ˆ ˆ ] [ ] [ ]} [ { -∞ = - - - -∞ = = = = m j m j m m j e X e m x e m x ) ( ] [ ] [ ˆ ) ˆ ( ˆ ϖ ϖ ϖ
Image of page 9

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
(vi) Frequency shifting (modulation) A discrete-time sequence multiplying with a complex sinusoid in the time domain has its DTFT shifted in frequency. ) ( ] [ ) ˆ ˆ ( ˆ 0 0 ϖ ϖ ϖ - j n j e X n x e Please do the proof of the above property and show that )] ( ) ( [ 2 1 ) ˆ cos( ] [ ) ˆ ˆ ( ) ˆ ˆ ( 0 0 0 ϖ ϖ ϖ ϖ ϖ + - + j j e X e X n n x (vii) Linear convolution The DTFT of the linear convolution of x 1 [ n ] and x 2 [ n ] in the time domain is the multiplication of the DTFTs of x 1 [ n ] and x 2 [ n ] in the frequency domain. ) ( ) ( ] [ ] [ ˆ 2 ˆ 1 2 1 ϖ ϖ j j e X e X n x n x Please read the proof in the textbook. (viii) Differentiation in frequency theorem The DTFT of a sequence ] [ ] [ n nx n y = is given by ϖ ϖ ϖ ˆ ) ( ) ( ˆ ˆ d e dX j e Y j j = ϖ ϖ ˆ ) ( ] [ ˆ d e dX j n nx j (ix) Autocorrelation sequence
Image of page 10
The autocorrelation sequence of a discrete-time signal x [ n ] is defined as -∞ = - = n xx l n x n x l r ] [ ] [ ] [ where the parameter l is called the lag. The autocorrelation sequence can be used to determine the periodicity of the signal. The autocorrelation sequence can be expressed as a convolution sum ] [ ] [ )] ( [ ] [ ] [ n x n x n l x n x l r n xx - = - - = -∞ = For l =0, -∞ = = n xx n x r 2 | ] [ | ] 0 [ is the energy of the discrete-time signal. If x [ n ] is a real-valued sequence and the DTFT of x [ n ] is X ( ϖ ˆ j e ), then the DTFT of the autocorrelation sequence is 2 ˆ ˆ ˆ ˆ | ) ( | ) ( ) ( ) ( ]} [ * ] [ { ϖ ϖ ϖ ϖ j j j l l j xx e X e X e X e l r n x n x DTFT = = = - - -∞ = - or - = π π ϖ ϖ ϖ π ˆ | ) ( | 2 1 ] [ ˆ 2 ˆ d e e X l r l j j xx The energy is equal to r xx [0] and can be expressed as - -∞ = = = = π π ϖ ϖ π ˆ | ) ( | 2 1 | ] [ | ] 0 [ 2 ˆ 2 d e X n x r E j n xx The above relation is called as Parseval’s Theorem and | X ( ϖ ˆ j e )| 2 is called as energy density spectrum.
Image of page 11

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Example: Find - + - π π ϖ ϖ ϖ α ˆ ˆ sin ) ˆ cos 1 ( 1 2 2 d We have ϖ ϖ α ϖ ˆ sin ) ˆ cos 1 ( 1 | ) ( | 2 2 2 ˆ + - = j e X where its discrete-time sequence is . 1 | | ], [ ] [ < = α α n u n x n Based on Parseval’s Theorem, we have 2 0 2 2 2 1 2 | | 2 ˆ ˆ sin ) ˆ cos 1 ( 1 α π α π ϖ ϖ ϖ α π π - = = + - = - n n d Frequency Response The DTFT of the impulse response of an LTI system is called as the Frequency Response of the system -∞ = - = n n j j e n h e H ϖ ϖ ˆ ˆ ] [ ) ( The DTFT of the output of an LTI system with the frequency response H ( ϖ ˆ j e ) and the DTFT of the input, X ( ϖ ˆ j e ), is given by ) ( ) ( ) ( ˆ ˆ ˆ ϖ
Image of page 12
Image of page 13
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern