The output of an N-point FFT (Fast Fourier Transform) is zero at odd positions if its inputs at odd positions with N/2 distance are equal and this property is exploited in order to reduce the power consumed by the FFT circuitry when operating on sparse data. An OFDM (Orthogonal Frequency Division Multiplexing) transceiver with such a configurable FFT is described, where undersampling is applied in order to reduce the ADC (Analog Digital Conversion) power and the buffering requirements by up to 25% or even 37.5% if additional FFT properties are exploited. As will be shown in this paper, up to 3/8 of the FFT butterfly operations can be deactivated in this case without any effect on the error caused by the undersampling procedure. Deactivating a higher number of FFT operations is also possible with a small error overhead. The reconstruction of the sparse images captured by a surveillance camera as well as X-ray images is also used as case studies demonstrating the use of the proposed method in image transfer applications over OFDM infrastructures. The Normalized Mean Square Error (NMSE) in these case studies is as low as 0.01 if 12.5% of the FFT input samples are omitted by the undersampling process.