# Phase function is an odd function of ϖ ˆ making use

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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

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(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
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.

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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
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