10-1 Discrete-Time Fourier Transform (伅聽葚鉥?

Old Chinese version

Slides

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

• 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.

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:

X(ejw)	=	Ske-jwk x[k]
x[n]	=	(2p)-1∫2pX(ejw)ejwn dw

We can derive the second expression from the first expression easily. Since the period of X(e^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:

∫02p f(x) dx = limN→∞Sk=0N-1 f(k•(2p/N))•(2p/N)

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

x[n]	=	Re{(2p)-1∫2pX(ejw)ejwn dw}
	=	(2p)-1∫2p Re{X(ejw)ejwn} dw
	=	(2p)-1∫2p |X(ejw)| cos(wn + q) dw, q=∠X(ejw)
	=	N-1 Sk=0N-1 [|X(ejw)| cos(wn + q)]w=2pk/N dw, 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^jw)|/N.

From the discussion in this section and the previous section, we know that there are two important meanings of DTFT：

1. If h[n] is the impulse response of an LTI system, then H(e^jw)=S_kh[k]e^-jwk represents the gain (|H(e^jw)|) and phase shift (∠H(e^jw)) when the input signal's angular frequency is w.
2. If x[n] is an arbitrary signal, then X(e^jw)=S_kx[k]e^-jwk represents the amplitude (|H(e^jw)|) and the phase shift ( ∠H(e^jw)) of the component at w of x[n].

Some important properties of DTFT are listed next.

1. Linearity: z[n] = a x[n] + b y[n] ←→ Z(e^jw) = a X(e^jw) + b Y(e^jw)
2. Periodicity: X(e^{j(w + 2 k p)}) = X(e^jw) Namely, the period of DTFT is 2p。
3. Delay in time: y[n] = x[n-k] ←→ Y(e^jw) = e^jwk X(e^jw)
4. Convolution: y[n] = x[n]*h[n] ←→ Z(e^jw) = X(e^jw)Y(e^jw) That is, convolution in time domain corresponds to multiplication in frequency domain.

If x[n] is a real sequence, we can separate its DTFT X(e^jw) into the real and imaginary parts:

X(ejw)	=	Ske-jwk x[k]
	=	Skx[k] cos(wk) - j Skx[k] sin(wk)
	=	XR(ejw) + j XR(ejw)

其中 X_R(e^jw) = S_kx[k] cos(wk)
X_I(e^jw) = - S_kx[k] sin(wk) This functions have the following properties:

• 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.
• ∠X(e^jw) is an odd function.

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