Traditionally,such analysis is carried out by spectrum analyzers which were banks of filters. We will see later in this lecture how a filter bank carries out this task.Next time we will incorporate the FFT into the filter bank implementation. The Time-Dependent Fourier Transform Also known as the short-time Fourier transform(STFT),the time-dependent Fourier transform (TDFT)of a signal x[n]is defined as xn,)=∑xn+miwimle-m, where wln]is a window sequence.As time progresses,the signal slides past the fixed window. Here we use A as the frequency variable to distinguish from the conventional DTFT,and the mixed bracket-parenthesis notation Xn,A)to notate that n is a discrete variable while A is a continuous variable. As an illustration of how the window wn is positioned relative to the shifted signal,OSB Figure 10.11 shows a linear chirp signal with the window superimposed.The signal being shifted through time is x[n]=cos(won2), w%=2π×7.5×10-6 Its instantaneous frequency is 2won (hence the name 'linear chirp').w[n]used is a Hamming window of length 400.Over the finite duration of the window,the frequency content of the signal changes only slightly by 400wo =27(0.003).We can see the linearity of the signal's frequency content from OSB Figure 10.12,a STFT magnitude plot where the vertical axis represents continuous frequency A and the horizontal axis represents discrete time n.Such a magnitude plot is referred to as a spectrogram. The Effect of the Window From the definition of STFT,we see that x)-DTFTmDTFTDTFTfumj) For a fixed n, 12m Xm,0=2示0 eionx(eio)w(ei(-0))do This is the convolution between the window W(e)and the shifted input sequence.The main- lobe width of W(ew)determines how well the components of the signal frequency spectrum can be resolved and the sidelobe amplitude determines the amount of leakage from these compo- nents into their neighbors.Again,trade-offs exist when selecting a proper window for short-time Fourier analysis(STFA).OSB Table 7.1 in Section 7.2 quantitatively illustrates such trade-offs 2Traditionally, such analysis is carried out by spectrum analyzers which were banks of filters. We will see later in this lecture how a filter bank carries out this task. Next time we will incorporate the FFT into the filter bank implementation. The Time-Dependent Fourier Transform Also known as the short-time Fourier transform (STFT), the time-dependent Fourier transform (TDFT) of a signal x[n] is defined as ∞ e X −jλm [n, λ) = � x[n + m]w[m] , m=−∞ where w[n] is a window sequence. As time progresses, the signal slides past the fixed window. Here we use λ as the frequency variable to distinguish from the conventional DTFT, and the mixed bracket-parenthesis notation X[n, λ) to notate that n is a discrete variable while λ is a continuous variable. As an illustration of how the window w[n] is positioned relative to the shifted signal, OSB Figure 10.11 shows a linear chirp signal with the window superimposed. The signal being shifted through time is x[n] = cos(ωon2), ωo = 2π × 7.5 × 10−6 . Its instantaneous frequency is 2ωon (hence the name ‘linear chirp’). w[n] used is a Hamming window of length 400. Over the finite duration of the window, the frequency content of the signal changes only slightly by 400ωo = 2π(0.003). We can see the linearity of the signal’s frequency content from OSB Figure 10.12, a STFT magnitude plot where the vertical axis represents continuous frequency λ and the horizontal axis represents discrete time n. Such a magnitude plot is referred to as a spectrogram. The Effect of the Window From the definition of STFT, we see that 1 X[n, λ) = DTFT{x[n + m]w[m] DTFT } = {x[n + m]} ∗ DTFT{w[m] 2π } For a fixed n, 1 � 2π X[n, λ) = ejθnX(ejθ)W(ej(λ−θ) )dθ . 2π 0 This is the convolution between the window W(ejω) and the shifted input sequence. The main￾lobe width of W(ejω) determines how well the components of the signal frequency spectrum can be resolved and the sidelobe amplitude determines the amount of leakage from these compo￾nents into their neighbors. Again, trade-offs exist when selecting a proper window for short-time Fourier analysis (STFA). OSB Table 7.1 in Section 7.2 quantitatively illustrates such trade-offs. 2