In the previous chapter, we have covered several important concepts:

In particular, for a given discrete-time signal x[n], its discrete-time Fourier transform (DTFT for short) and the corresponding inverse transform can be expressed as follows:

- The frequency response of an LTI system is the discrete-time Fourier transform of the system's impulse response.
- The frequency response of an LTI system specifies how the system changes the amplitude and phase of an input sinusoidal function with a specific angular frequency w.
We can derive the second expression from the first expression easily. Since the period of X(e

X(e ^{jw})= S _{k}e^{-jwk}x[k]x[n] = (2p) ^{-1}¡ì_{2p}X(e^{jw})e^{jwn}dw^{jw}) is 2p, we can use any interval of length 2p for integration.From the second expression, we can decompose x[n] into an infinite linear combination of e

^{jwn}, where the amplitude of each basis function is specified by X(e^{jw}). In other words, X(e^{jw}) is actually the amplitude of the component of x[n] at the continuous angle frequency w. This can be futher elaborated, as follows.From elementary calculus, we know that the integration of f(x) can be approximated by summation:

Therefore if x[n] is real, we can approximate it by using summation:

¡ì _{0}^{2p}f(x) dx = lim_{N¡÷¡Û}S_{k=0}^{N-1}f(k¡E(2p/N))¡E(2p/N)From the derivation above, x[n] has been decomposed into the linear combination N cosine functions with the angular frequency w from 0 to 2p(N-1)/N, and the corresponding amplitude |X(e

x[n] = Re{(2p) ^{-1}¡ì_{2p}X(e^{jw})e^{jwn}dw}= (2p) ^{-1}¡ì_{2p}Re{X(e^{jw})e^{jwn}} dw= (2p) ^{-1}¡ì_{2p}|X(e^{jw})| cos(wn + q) dw, q=¡çX(e^{jw})= N ^{-1}S_{k=0}^{N-1}[|X(e^{jw})| cos(wn + q)]_{w=2pk/N}dw, N¡÷¡Û^{jw})|/N.From the discussion in this section and the previous section, we know that there are two important meanings of DTFT¡G

- If h[n] is the impulse response of an LTI system, then H(e
^{jw})=S_{k}h[k]e^{-jwk}represents the gain (|H(e^{jw})|) and phase shift (¡çH(e^{jw})) when the input signal's angular frequency is w.- If x[n] is an arbitrary signal, then X(e
^{jw})=S_{k}x[k]e^{-jwk}represents the amplitude (|H(e^{jw})|) and the phase shift ( ¡çH(e^{jw})) of the component at w of x[n].Some important properties of DTFT are listed next.

If x[n] is a real sequence, we can separate its DTFT X(e

- Linearity:
z[n] = a x[n] + b y[n] ¡ö¡÷ Z(e ^{jw}) = a X(e^{jw}) + b Y(e^{jw})- Periodicity:
X(e Namely, the period of DTFT is 2p¡C^{j(w + 2 k p)}) = X(e^{jw})- Delay in time:
y[n] = x[n-k] ¡ö¡÷ Y(e ^{jw}) = e^{jwk}X(e^{jw})- Convolution:
y[n] = x[n]*h[n] ¡ö¡÷ Z(e That is, convolution in time domain corresponds to multiplication in frequency domain.^{jw}) = X(e^{jw})Y(e^{jw})^{jw}) into the real and imaginary parts:¨ä¤¤

X(e ^{jw})= S _{k}e^{-jwk}x[k]= S _{k}x[k] cos(wk) - j S_{k}x[k] sin(wk)= X _{R}(e^{jw}) + j X_{R}(e^{jw})X This functions have the following properties:_{R}(e^{jw}) = S_{k}x[k] cos(wk)

X_{I}(e^{jw}) = - S_{k}x[k] sin(wk)

- Conjugate symmetric: X
^{*}(e^{jw}) = X(e^{-jw})- X
_{R}(e^{jw}) is an even function.- X
_{I}(e^{jw}) is an odd function.- |X(e
^{jw})| is an even function.- ¡çX(e
^{jw}) is an odd function.

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