... bewritten asX k X1 k W k NX2 k ; k 0, 1, , N À 1X1 k W k NX2 k , k 0, 1, , N=2À1X1 k ÀW k NX2 k , k N=2, , N À 1,@7:2:6where X1 k DFTx1m and X2 k DFTx2m ... should be used?11. A 1 kHz sinusoid is sampled at 8 kHz. The 128-point FFT is performed to compute X (k) . Atwhat frequency indices k we expect to observe any peaks in jX k j?12. A touch-tone phone ... ancncÃn=2 and bnÀjcnÀcÃn=2, then they can be written in termsof DFTs as and A k C k CÃN À k =2 and B k ÀjC k ÀCÃN À k =2.Finally, the real input FFT can be obtained by X k A k W k 2NB k ...