## 10-1 Discrete-Time Fourier Transform (嚙踝蕭嚙踝蕭嚙踝蕭嚙踝蕭嚙踝蕭嚙踝蕭嚙踝蕭嚙踝蕭嚙踝蕭嚙踝蕭嚙踝蕭嚙踝蕭嚙踝蕭嚙?

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(ejw) 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 ejwn, where the amplitude of each basis function is specified by X(ejw). In other words, X(ejw) 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(ejw)|/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(ejw)=Skh[k]e-jwk represents the gain (|H(ejw)|) and phase shift (∠H(ejw)) when the input signal's angular frequency is w.
2. If x[n] is an arbitrary signal, then X(ejw)=Skx[k]e-jwk represents the amplitude (|H(ejw)|) and the phase shift ( ∠H(ejw)) 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(ejw) = a X(ejw) + b Y(ejw)
2. Periodicity:
X(ej(w + 2 k p)) = X(ejw)
Namely, the period of DTFT is 2p
3. Delay in time:
y[n] = x[n-k] ←→ Y(ejw) = ejwk X(ejw)
4. Convolution:
y[n] = x[n]*h[n] ←→ Z(ejw) = X(ejw)Y(ejw)
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(ejw) 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)

XR(ejw) = Skx[k] cos(wk)
XI(ejw) = - Skx[k] sin(wk)
This functions have the following properties:
• Conjugate symmetric: X*(ejw) = X(e-jw)
• XR(ejw) is an even function.
• XI(ejw) is an odd function.
• |X(ejw)| is an even function.
• ∠X(ejw) is an odd function.

Audio Signal Processing and Recognition (音訊處理與辨識) 